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Curve Sketching Derivatives

TopHere we have to create or sketch a graph using the Derivatives or differentiation. The process to create or sketch is fairly simple. The process of sketching graph is shown in steps below-
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 + 3x2 - 45x?
Solution ) Find f ' (x) we get f' (x) = 3x2 + 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.