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Calculus Absolute Extrema

TopIf we have a function f(x) which is continuous in an interval [a, b] and we want to calculate the absolute extrema, we have to follow some steps which are as follows:
  1. First, we have to check whether a function is continuous or not in an interval [a, b].
  2. Then, we have to find out all the critical points of function f(x) in the interval [a, b]. Here, we only observe the critical points in a given interval, where the function is continuous for any other interval.
  3. Then, we have to find out the function where the critical points are found.
  4. Finally, we have to identify the absolute extrema.

To understand this more deeply, we just take an example of any function.
k(x) = 2x3 + 3x2 - 12x + 4 on [-4, 2]

Here, we have to find out the absolute extrema for a given function and interval. We can easily see that, this is a polynomial equation. So, it is continuous in each interval, especially in [-4, 2]. Now, we have to find the critical points of the function. Thatswhy, we find the derivative of the above function which is,
k'(x) = 6x2 + 6x - 12

Then, we separate the equation in factors as follows:
k'(x) = 6 (x + 2) (x - 1)

So, after solving further, we find that x = -2 and x = 1 as the two critical points. Now, we have to calculate the end points of interval and the function at the critical points. So,
K(-2) = 24k(1) = -3
K(-4) = -28k(2) = 8

There are four points, where the calculus absolute extrema can occur. But, according to absolute extrema, it is the largest and the smallest of function. So,
  • Absolute maximum k(x) = 24 occurs at x = -2.
  • Absolute minimum k(x) = -28 occurs at x = -4.