Step 1 ) First step is to find the derivative of function that is f'( x ) and if f' (x) > 0 then on interval ' I ' than function is increasing on interval ' I ' and if f' (x) < 0 than function is decreasing.

Step 2 ) Now find the second derivative of the function that is f'' (x).

Step 3 ) Put f'' (x) equals to 0 we get f''(x) = 0.

Step 4 ) If f'' (x) > 0 than function f(x) is concave up.

Step 5 ) If f'' (x) < 0 than function f(x) is concave down.

Step 6 ) Now find relative extrema by putting values of ' x '.

Step 7 ) Calculate the Point of inflection by putting x = a.

Let us take an example to

Curve sketching derivatives, the example is shown below-

Example ) Sketch curve for function f(x) = x

^{3}+ 3x

^{2}- 45x?

Solution ) Find f ' (x) we get f' (x) = 3x

^{2}+ 6x - 45.

Now put f' (x) = 0 we get x = 3 and x = -5, therefore f(x) is increasing on

f ( -∞, -5 ), decreasing on ( -5 , -3 ), increasing on ( 3 , ∞ ).

The graph has local maximum at ( -5 , f ( -5 ) ) = ( -5 , 175 ) and ( 3, f(3) ) = ( 3 , -81 ) is local minima of graph.

Now find f''(x) we get f''(x) = 6x + 6, Put f'' (x) = 0, we get x = -1 so f(x) has concave up on ( -∞ , -1 ) and concave down on ( -1 , ∞ ).

The inflection point for this graph is ( -1 , f ( -1 ) ) = ( -1 , 47 ) .

Now graph will look like.