Maxima and minima of a function together can be called as extrema. Maxima and minima can be defined as largest and smallest of a function at a given Point in its Domain or outside its domain. We can calculate the maxima and minima of a function by using maxima and minima Calculus. To calculate maxima and minima we should follow the steps below- 1 ) Calculate f' ( x ). 2 ) Now putting f' ( x ) = 0 , we will get two values of ' x '. 3 ) We will now calculate f'' ( x ) and put first value of ' x ' in f'' ( x ). 4 ) If value of f'' ( x ) is positive than function is minimum otherwise function is maximum. |

The maximum and minimum values of a function collectively known as extrema .These are the smallest and largest values that a function takes at a Point in a given neighborhood local or relative extrema or on its Domain which is entirety global or absolute extrema. The Set of unbounded infinite real Numbers have no minimum and maximum. Now the analytical definition of the function, the real valued function for local or relative maximum point at the point x^{n}, if there exists ε > 0 and such that f(x^{n}) ≥ f(x) when |x – x^{n}| < ε. Such value of the function at that point is known as maximum of the function.

On other hand, if f (x^{n}) ≤ f(x) and |x – x^{n}| < ε then a function has a local minimum point at x^{n}. The value of the function at that point is known as minimum of the function. If a function f f (x^{n}) ≥ f(x) for all x then the function has a global or absolute maximum point at x^{n}. Similarly, if f(x^{n}) ≤ f(x) for all x then a function has a global or absolute minimum point at x^{n}. The other names of global maximum and global minimum points are the arg max and arg min. These are the argument input at which the maximum or minimum occurs. There is a domain which is known as restricted domains and defined as restricted domains are those Functions whose domain does not include all Real Numbers.

A function whose domain is any set, can have a global maximum and minimum is a real-valued function. We can find the value of maximum and minimum of a continuous function whose domain is a closed and bounded interval of real numbers. The endpoint of any bounded interval will be a global maximum or minimum. Note one thing that it is not always true that if the function has global maximum minimum then it is not necessary that they have a local maximum minimum value.

Now we find global Maxima and Minima when the extreme value theorem of global maxima and minima exist then a function is continuous on a closed interval. In the internal domain we can say that the value of local maximum or minimum is also the value of global maximum or minimum of its domain. When we are finding the global maximum or minimum methods we find the all the local maxima or minima and then also take the small or big values. There is a theorem which helps to find the local extrema and the theorem is Fermat's theorem. We can find a critical point but firstly if we have local maximum or local minimum values by using the first derivative test or second derivative test. Now we are discussing some definitions which are related to the maximum and minimum values of a function in its domain,

Function of two variables has a local maximum at point (x, y) if f(x, y) ≤ f(x, y) for all points (x, y) in center (x, y).The number f(x, y) called a local maximum value. If f(x, y) ≥ f(a, b) for all (x, y) then f(x, y) is a local minimum value. A

(a, b) if f(x, y) ≤ f(a, b) for all points (x, y) in domain of function f function of two variables has a absolute maximum and (a, b) if f(x, y) ≥ f(a, b) for all points (x, y) in domain of function of two variables has a absolute minimum value. Let a function f(x, y) be a smooth function. We represent the domain of this function as D. Let the point I = (a**0**, b0) be in D. The point ‘I^{’} is called INTERIOR point and if there exists a disk centered at “I” then this disk is in D. The point “I” is called BOUNDARY POINT if “I” has a point which does not belong to D.

Let’s take a function f(x, y) and the point is (m, n) of local maximum or minimum. Assume that the function is continuous. So that this shows the ﬁrst order Derivatives fx, fy. The point C = (x0, y0) called the critical point of the function f if ∇f(x0, y0) = 0. Now we take an example to understand the above discussion,

Let’s take a function f(x, y =y= x^{3} +4x^{2} -8x+17 now firstly we find the first derivative of given function that is

dy/dx = 3x^{2} +10x +8.Equate the above function to 0

dy/dx =0, so we get 3x^{2} +10x +8=0

Now we factories the equation so obtained to find critical points

3x^{2} +6x+4x +8=0, we get factors as

3x(x+2) + 4(x+2) =0,(3x+4)(x+2)=0

Now we find the critical points

3x+4 =0 and x+2 =0 gives x= -4/3 and -2.

From the definition of minima and maxima if f "(x) >0 then given point is minima and If f "(x) < 0 then given point is maxima.

Now we find second derivative of the function

d^{2} y/dx^{2} =6x+10

Plug in critical values of x in second derivative.

d^{2} y/dx^{2} = 6(-4/3)+10= 2 and d^{2} y/dx^{2} = 6(-2)+10=-2

Therefore the values are,

x= -4/3 d^{2} y/dx^{2} >0 thus minima exists at this point. In this way will find the critical points and maximum and minimum values of a function. Lets we take another example which shows the unique minimum value function that isy=x^{2} the first derivative of given function is f^{’}(x) =2x.If plugging 2x=0 then x=0. And the second derivative is

f^{’’}(x)=2.So that the function *x*^{2} has a unique minimum at *x* = 0.

Now in case of a partial order, the smaller element than all other element should not be confused with a minimal element that is a greatest element of a partially ordered set is upper bound of the set which contain within the set. On other hand a maximal element n of a set S is an element of S such that if n ≤ b then n = b...This is all about Maximum and Minimum values of a function in its domain.

Local maxima is the Point that is at the peak with respect to the local surrounding points. Similarly, local minima can be defined as the lowest point with respect to the surrounding points. Maxima or minima can be many points as they are considered locally in the graph. It also resembles with the Set theory in a context that we have least and peak points in the set.A function g ( x ) has a maximum point locally at some point ‘x

_{max}’ if we have,µ > 0 for someµ.

Such that g ( x

_{max}) > = g ( x ) if and only if,

| x – x

_{max}| < µ.

If we find the value of the function at the given point, then it is called as the maximum point.

Similarly, local minimum points can be defined.

A function g ( x ) is said to have a local minimum point at ‘x

_{min}’ if,g ( x

_{min}) < = g ( x ) for all values of ‘x’ under the Domain, these are the global Maxima and Minima points. These are also known as ‘arg max’ and ‘arg min’ points.

Global maxima and minima are different from the local maxima and minima as they are taken globally and they are the highest or lowest points in the whole domain set.

While local maxima and minima are compared with the neighborhood points.

We have various methods to find the local maxima and local minima. Steps to find local maxima and minima:

1) The function is first differentiated with respect to the independent variable whose domain is described.

2) The first order derivative is then equated to zero in order to find the points of local maxima and local minima.

3) We have got the points, now we will check for the maximum and minimum points.

It can be checked by two methods which are described as follows,

Method 1: There are two conditions

Condition 1: If the graph goes from + ve to – ve at the point,it is a local maxima point.

Condition 2: If the graph is going from – ve to + ve , the point is said to be local minimum point.

Method 2: We will again differentiate the equation.

We have got the second order derivative of the function.

Our next step is to substitute the points in the second order derivative equation.

If the result is a ‘+ ve’ number, the point is called as local minima.

If result is ‘– ve’ number, the point is called as local maxima.

The second method is more efficient in many questions but first method is also used in some cases.

Theorem 1: If g (x) is a differentiable function that is defined in the interval ‘H’,if ‘a’ is a point in the interval ‘H’, then

1) ‘x = a’ is said to be a local maximum point if the first order derivative of ‘g ( x )’ is equals to zero at ‘x = a’, that is g ' ( a ) = 0 .

And the second condition is,

2 ) g ' ( x ) is changing its sign from ‘+ ve’ to ‘– ve’ as ‘x’ is increasing from ‘a’ , and g ' ( x ) > 0 at each point that is close to the right of ‘a’.

In mathematical terms, we have a function g (x).

step 1 ) Find g ' ( x ).

step 2 ) Equate g '( x ) to zero .

Step 3 ) g' ( x ) = 0, find g '' ( x).

step 4 ) : If g ''( m ) > 0,

Then function has a local minimum at x = m,

If g '' ( m ) < 0,

the function has a local maximum at x = m.

Let us discuss now some properties of local maxima and local minima.

Properties :

1 ) If we have a function g ( x ) defined in its domain and g ( x ) is also continuous, then there must be one maxima and minima point between two equal values of ‘x’.

2 ) There must be a maxima between the two minima points and vice versa. That is, there must be ‘a’ minima between two maxima points.

3 ) If we have a function g ( x) that is tends to infinity mathematically,

g ( x ) - > ∞,

When ‘x’ tends to ‘m’ or ‘n’, that is x - > m or n,and the first order derivative of g ( x ) is equal to zero.

g ' ( x ) = 0 for only one value of x = b, ‘b’ must lie within ‘m’ and ‘n’ that is m < b < n, then, f ( b ) is the minimum value.

If g ( x ) is tending to infinity,g ( x) - > ∞when ‘x’ is tending to ‘m’ or ‘n’, that isx - > m or n

and m < b < n then , there must be a point ‘f ( b )’ is said to be the maximum value at ‘x = b’ .

Maximum and minimum values are also defined in a closed interval.

Let us take a function g ( x ) which is defined in its domain [ m , n ].

If we have a local maximum or local minimum value at ‘x = b’ in the interval [m, n ] then, it implies that the minimum or maximum values exists in the neighbor of ‘b’.

A function can have many local maximum and minimum values but we can have only one global maximum and minimum value.Even a local maximum in one interval is greater than local maximum in another interval.

Local maxima of a function can be define as a function z=f(x) is said to be attains a local maximum value at x=a if there exist a neighborhood of ‘a’ such that f(x)<f(a)for all x belongs to the (neighborhood of ‘a’) and x is not equals to ‘a’. in such a way ,f(a) is known as local maximum value of f(x) at x=a.

Now let us define local minimum of a function; Local minimum of a function can be define as a function z=f(x) is said to be attains a local minimum value at x=a if there exists a neighborhood of ‘a’ such that f(x)>f(a) for all x belongs to the (neighborhood of ‘a’) an x is not equal to ‘a’. in such a way ,f(a)is known as local minimum value of f(x) at x=a.

The Point x=a where the function z=f(x) attains either local maximum or local minimum values are also known as extreme points or turning points and the local maximum and local minimum values are known as extreme values of the function z=f(x).

We should remember that the local maximum or local minimum of a function z=f(x) at a point x=a means the greatest or the lowest value in the neighborhood of the point x=a not the maximum or minimum value at the Domain of a function z=f(x)

For finding the local maximum or local minimum of a function z=f(x) we have to first find out the extreme point of the function.

The extreme point of a function can be obtain by putting dz/dx = 0 or we can say that by putting f’(x)=0 but in some cases if f’(a)=0 does not necessarily implies that x=a is a extreme point for example f(x) =x

^{3}implies that f’(0)=0 but at x=0 does not contain the extreme value.

For finding local maxima or local minima of a function z=f(x) there is a quite simple way which is also known as first order derivative test for Local Maxima and local Minima. First order derivative test for local maxima and local minima of a function: the first order derivative test for local Maxima and Minima can be define as let f(x) be a function define on an interval I and let suppose there is appoint ‘a’ belongs to the interval I then,

1: For x=a is a point of local maxima of the function f(x) if f’(a) = 0 and f’(x) changes its sign from positive to negative value as the value of x increases through ‘a’.

2: For x=a is a point of local minima of the function f(x) if f’(a) =0 and f’(x) changes its sign from negative to positive value as the value of x increases through ‘a’.

3: if f’(a) =0 and f’(x) does not changes its sign as x increases through ‘a’ means f’(x) have same sign in every neighborhood of ‘a’. Then x=a is known as point of inflexion.

Mathematically, we explain the first order derivative test for Local Maxima And Minima in following way, if there is a function z= f(x).

Find dz/dx and put dz/dx =0 and solve the equation for x?

Let a1, a2, a3, a4...are the roots of the equation determined by solving dz/dx=0.

Consider each point for x such as for x=a1 put in dz/dx . if dz/dx changes its sign from positive to negative as x is increases through a1 ,then the function attains local maximum at x= a1. If the value of dz/dx changes its sign from negative to positive as x increases through a1 then the function attains local minimum at x=a1.

If the value of dz/dx does not changes its sign as x increases through a1 then the function attains neither maximum nor minimum and the point x=a1 is called point of inflexion. Always remember that if the value of x for which f’(x) = 0 or f’(x) does not exist are known as critical points..

Example: find all the point of local maxima and local minima as well as the corresponding local maxima and local minima values of the function

f(x) = (x-1)

^{3}(x+1)

^{2}?

Solution: let us assume that y = f(x) = (x-1)

^{3}(x+1)

^{2}

For finding all the point of local maxima and local minima as well as the corresponding local maxima and local minima values we apply the first order derivative test for local maxima and minima. According to the first derivative test we will differentiate the given function with respect to x,

Therefore by the method of Product rule of differentiation

y’= f’(x) = 3(x-1)

^{2}(x+1)

^{2 }+ 2(x-1)

^{3}(x+1),

y = f’(x) = (x-1)

^{2}(x+1) (3(x+1) + 2(x-1)),

y = f’(x) = (x-1)

^{2}(x+1) (5x+1),

According to the first derivative test for local maxima or local minima of the function, we have

dy/dx = 0,

(x-1)

^{2}(x+1) (5x+1)= 0,

x = 1 or x = -1 or x = -1/5,

Now, we have to examine whether these values of x are the points of local maximum or local minimum or neither of them. Since (x-1)

^{2}is always positive for any value of x ,therefore the sign of dy/dx is same as the sign of (x+1) (5x+1). Now, we check at what value of x the sign of derivative dy/dx of the given function will change.

Clearly dy/dx does not change its sign as x passes through the point x= 1.

Therefore x = 1 is neither appoint of local minima nor a point of local maxima.

As well as at x=1 there is no local maximum or minimum value of the function .therefore we can say that x=1 is a point of inflexion.

Clearly again dy/dx change its sign from positive to negative as x passes through the point x= -1.

Therefore x=-1 is a point of local maxima. of the given function and the local maximum value of the function at x=-1 will be:

f(-1)= (-1-1)

^{3}(-1+1)

^{2}

f(-1)= (-2)

^{3}(0) = 0

Therefore at x=-1 is a point of local maxima of the given function and the local maximum is equals to 0

Clearly again dy/dx change its sign from negative to positive as x passes through the point x= -1/5.

Therefore x=-1/5 is a point of local minima. of the given function and the local minimum value of the function at x=-1/5 will be:

f(-1/5)= (-1/5-1)

^{3}(-1/5+1)

^{2}

f(-1)= (-6/5)

^{3}(4/5) = -3456/3125

Therefore at x=-1/5 is a point of local minima .of the given function and the local minimum value is equals to -3456/3125.

Using the above mentioned step and process you can easily solve different problems related to local maxima and minima.

The higher order derivative in the Calculus mathematics is a technique to find the Point of inflection in the function. Generally, it is used in the calculus for the purpose of finding the maxim’s and minima's of a time differentiable and time varying Functions.

Higher order Derivatives are the type of derivatives which are found by performing several times differentiation of a differentiable function. For example, say “f” is a function in x, which is n time differentiable and can be assumed in the form of:

f ( x ) = a_{0} x^{n} + a_{1} x ^{(n-1)} + a_{2} x ^{(n-2)} + a_{3} x ^{(n-3)} + ….......... + a_{(n-2)} x^{2} + a_{(n-1)} x^{1} + a_{(n)} x^{0}

Now, from the equation f ( x ) we can see that, this is a function of x, which is n times differentiable and for this function we can calculate the n times higher order derivatives of this function. To find the first order derivative of the function f (x ), we have to differentiate it one time. So, first derivative f' (x) will be:

f' (x) = a_{0} n x^{(n-1 )}+ a_{1} (n-1) x ^{(n-2)} + a_{2} (n-2) x ^{(n-3)} + a_{3} (n-3) x ^{(n-4)} + ….......... + a_{(n-2)} 2x + a_{(n-1)} x^{0}+ a_{(n)} * 0.

f' (x) is the first order derivative of the function f (x), and from the first order derivative equation we can again say that it is a differentiable function and again can be differentiated in to its next lower order differentiations. The next derivative of the function f'(x) is called as the second order derivative of the function and given as f''(x) or f^{2}(x). To find the second order derivative f'' (x) of the function we have to do differentiation of this function again.

f'' (x) = a_{0} n (n-1)x^{(n-2)}+ a_{1} (n-1)(n-2) x ^{(n-3)} + a_{2} (n-2)(n-3) x ^{(n-4)} + a_{3} (n-3)(n-4) x ^{(n-5)} +….......... + a_{(n-2)} 2.

Similarly, we can calculate the next lower derivatives called third order derivative, fourth order derivative, and so on. The differentiation process continued to its lower order derivatives to find all the successive derivatives of the function. To find a n^{th} order derivative of a function we have to do differentiation of the function as many times as the last term of the differentiation becomes equal to the zero. The n^{th} derivative of the function is given by f^{(n)}(x). So the n^{th} derivative of the function f^{(n)}(x) can be given as:

Let's see an example of the higher order equation:

Find the first, second, and third derivatives of f( x) = x^{4} − 3x^{3} + 5x^{2} − 7x + 8.

Now, if we go through the solution of the question, then we have to do several differentiation of the differentiable expression. The first differentiation of this can be given as:

f'( x) = 4x^{3} – 3( 3x^{2} ) + 5 * (2x) − 7

f'( x) = 4x^{3} – 9x^{2} + 10x − 7

As the f'(x) is also a function of x so we can find the second derivative of the function and it can be find by differentiating the first order differentiation equation. So, the second order derivative f''(x) of the function f(x) will be:

f''( x) = 4 * (3x^{2}) – 9( 2x ) + 5 * 2

f''( x) = 12x^{2} – 18x + 10

Similarly the third order, fourth order derivative of the function can be given as:

f'''(x) = (12x2) x – 18

f^{3}(x) = 24 x – 18

And the fourth order derivative of the equation will be:

f^{4}(x) = 24 – 18 * 0

f^{4}(x) = 24.

Now, talk about what is logical explanation of the higher order derivative test and also about the fact that why we keeps on the differentiation process of any function again and again. We do this because of finding the minima and the maxima of any function that where the values of the function are maximum and where they are at their minimum value strength.

We take the first derivative of the function and Set it to zero, and after that we solves the f'(x) for some values of x, for some of the values of the x there will be some maximum and some of the minimum values of that function and while Graphing these values of the function, we get that curve of the first derivative is somewhere maximum and somewhere minimum and the Slope of the curve at these points are zero.

The next test what we performs is that, we take the second derivative of the function and we see, if the second derivative f''(x) is positive or negative at x. The positive value of second derivative gives the minimum and the negative value of the function f''(x) gives the maximum value of the original function.

So, first derivative of the function simply shows the change in the rate at which the first derivative changes. The second derivative shows the change in the Slope of first derivative.

Here are several facts regarding the maximum and minimum values of the function. Say the n^{th} order derivative of the function f(x) is f^{n}(x) and if f^{n}(a) exists and it is non zero value then we can say following:

1. If the number n is a even number then:

a) f^{n}(x) < 0 which implies that the x = a is a point of local maximum.

b) f^{n}(x) > 0 which implies that the x = a is a point of local minimum.

2. If the number n is an odd number then:

a) f^{n}(x) < 0 which implies that the x = a is a strictly decreasing point of inflection.

b) f^{n}(x) > 0 which implies that the x = a is a strictly increasing point of inflection.

Inflection points can be defined as the points where a function has a change in its concavity. If second derivative of a function is positive, than it forms an upper concave, but if second derivative of a function is negative than it forms a down concave. The Point at which function changes the concave up to concave down or from concave down to concave up, at this point of change the second derivative must be equals to 0. So second derivative should be equals to 0 to be an inflection point. We must be sure that there is a change in concavity at that point.

Let’s take an example to understand the concept of points of inflection-

Suppose we have a function f(x) = x** ^{3}**, find the point of inflection for this function.

To solve this we need to first calculate the first derivative of the given function f(x) = x3, we get d(x3) = 3x2 so f' (x) = 3x2.

Now we will find the second derivative of the function f(x) we get f''(x) = 6x. Now put second derivative equals to 0. So here we get f''(x) = 0 that is 6x = 0, we get x = 0.

So the inflection point will e at x = 0, but we need to make sure that inflection point is at x = 0. To check this select a value on each side of ' x ', that is x = -1 and x = 1. Put x = -1 in second derivative, so we get f’’ (-1) = 6(-1) = -6.

Now put x = 1 in f''(x), we get f’’ (1) = 6(1) = 6.

Therefore the function is concave up at x = 1 and concave down x = -1.

This is how we calculate inflection points.

A function can take two typeof values that are classified as maxima and/or minima. The Maxima and Minima are also given a name, which is extrema. We can think of maxima and minima as the functional value which can be the highest and the lowest possibility of it. Extrema can be called as global or local. As the name directly indicates a global maximum is the maximum value which a function achieves, while the global minimum is the minimum value which a function achieves. If we take any bounded function, then the global maximum is the highest and the global minimum is the lowest value of the given function.

Following are some properties of maxima and minima: Both the maximum and minimum of a function must have a finite number. Therefore, if a function increases without bound, that function has no global maximum. Likewise, if a function decreases without bound, there is no global minimum.

The Maxima or minima value of any function plays a major role in our daily life applications. They help us every where we want to find the highest or the lowest possible value of any function.

Let us consider any graph say y = f(x). The graph has a local maximum at any Point where the graph starts changing from increasing to decreasing. We observe that at such point the Tangent of the given graph has Slope = 0. Also the graph has local minimum at any point where the graph starts changing from decreasing to increasing. We again observe that at such point the tangent of the given graph has Slope = 0. To find maxima or minima of a function first find its first derivative, solve the f’(x) =0 and find max or min. Then check the value is max or min by finding the nearby values and calculate the function for that value.

Let g(x) is a real function defined on an interval ‘I’. Then g(x) is said to have the maximum value in the interval ‘I’ if there exist a Point ‘c’ in the interval ‘I’ such that, g(x) ≤ g(c) for all ‘x’ belongs to ‘I’.In this case the number g(c) is called the maximum value of g(x) in the interval ‘I’ and the point ‘c’ is known as a point of maximum value of function ‘f’ in the interval ‘I’.

Let g(x) is a real function defined in an interval ‘I’. Then g(x) is said to have the minimum value in the interval ‘I’, if there exist a point ‘c’ in the interval ‘I’ such that, g(x) ≥ g(c) for all ‘x’ belongs to ‘I’.

Thus we conclude that for a function g(x) defined on an interval ‘I’ i.e.

A function may attain the maximum value at a point in the interval ‘I’ but not the minimum value at any point in the interval ‘I’.

A function may attain the minimum value at a point in the interval ‘I’ but not the maximum value at any point in the interval ‘I’.

A function may attain both maximum and minimum values at some points in the interval ‘I’.

A function may not attain any of the values in the interval ‘I’ i.e. neither maximum or nor minimum.

Here we discussed about the greatest or the least values of a function in its Domain but there may be some points in the domain of a function where the function does not attain the greatest or least value but the least values at these points are greater than or less than the values of the function at the neighboring points. Such points are known as global maxima or global minima and can be explained as-

A function g(x) is said to attain a local maximum at x = c, if there exists a neighborhood

(c – δ, c – δ) of ‘c’ such that, g(x) < g(c) for all ‘x’ belongs to (c – δ, c + δ), where ‘x’ is not equals to ‘c’.

A function g(x) is said to attain a local minimum at x = c if there exists a neighborhood

(c – δ, c – δ) of ‘c’ such that, g(x) > g(c) for all ‘x’ belongs to (c – δ, c + δ), where ‘x’ is not equals to ‘c’.

Theorem:

A necessary condition for g(c) to be an extreme value of a function g(x) is that g'(c) is equal to zero in the case it exists.

This indicates to the some remarks for the Maxima and Minima.

This result states that if the derivative exists, it must be zero at the extreme points. A function may attain an extreme value at a point without being derivable. For example the function g(x) = |x| attains the minimum value at the origin even though it is not derivable at x = 0.

This condition is only a necessary condition for the point x= c to be an extreme point, g'(c) = 0 does not necessarily imply that x = c is an extreme point. There are some Functions for which the Derivatives vanish at a point but do not have an extreme value. For example for the function g(x) = x

^{3}, g'(0) = 0 but at x = 0 the function does not attain an extreme value.

Geometrically the above condition means that the Tangent to the curve y = g(x) at a point where the Ordinate is maximum or minimum is parallel to the x- axis.

As discussed in the second point that all ‘x’ for which g'(x) = 0 do not give us the extreme values. The values of ‘x’ for which g'(x) = 0 are called stationary points or turning points and the corresponding values of g(x) are known as stationary values of g(x).

The values of ‘x’ for which g'(x) = 0 or g'(x) does not exist are known as critical points.

To calculate the maxima and minima of a function g(x) first order and higher order differentiable theorems are as follows.

First order differentiable method to find the local maximum and minimum values.

Suppose that ‘g’ is a differentiable function defined on an interval ‘I’. Let ‘c’ belongs to I then

x = c is a point of local maximum value of the function ‘g’, if

g'(c) = 0 and,

g'(x) changes sign from positive to negative as ‘x’ increases through ‘c’ that is g'(c) > 0 at every point sufficiently close to and to the left of ‘c’ and g'(x) < 0 at every point sufficiently close to and to the right of ‘c’.

x = c is a point of local minimum value of the function ‘g’, if

g'(c) = 0 and,

g'(x) sign changes from negative to the positive value as ‘x’ increases through ‘c’ that is g'(x) < 0 at every point sufficiently close to and to the left of ‘c’ and g'(x) > 0 at every point sufficiently close to and to the right of ‘c’.

if g'(c) = 0 and g'(x) does not change sign as in the increment in ‘c’ means g'(x) has the same sign in the complete neighborhood of ‘c’ then ‘c’ is neither a point of local maximum value nor a point of local minimum value, such a point is known as inflection point.

Higher order derivative test for finding Local Maxima And Minima,

Suppose that ‘g’ is a differentiable function on an interval ‘I’ and let a be an interior point of the interval ‘I’ such that-

g'(a) = g”(a) = g”'(a) = g””(a) = …. = g

^{n-1}(a) = 0,

g

^{n}(a) exists and is a non zero function.

Then

If ‘n’ is an even number and g

^{n}(a) < 0 then x = a will be a point of local maximum.

If ‘n’ is an even number and g

^{n}(a) > 0 then x = a will be a local minimum.

If ‘n’ is an odd number then x = a will neither be a point of local maximum nor a point of local minimum. This is how we find the local maximum and minimum values.