·       Sketching  Graph of Derived Function

 

At the stationary points, the value of f’(x) is always zero.

This means that the graph of f’(x) will cut the x axis whenever

f(x) is a maximum, minimum or point of inflection.

 

 

Text Box: To sketch the graph of a derived function 1. Look for stationary points. Mark the x-axis underneath these points. 2. Investigate the nature of these stationary points • If the gradient is increasing, value of f’(x) is positive and will appear above the x-axis. • If the gradient is decreasing, value of f’(x) is negative and will appear below the x-axis. Mark points on graph and lightly sketch shape.

 

 

Example

 

 

 

 

 

 

x<a

x =a

x > a

f’(x)

+

0

+

 

 

 

Example

 

The graph of a function f intersects the x-axis at

(-a, 0), (h,0) and ( k,0)  and the y-axis at ( 0,d) as shown.

 

There are minimum turning points at ( -b, c) and ( i,j)

and  a maximum turning point at ( f, g).

 

Sketch the graph of the derived function  f’

 

 

 

 

Inspecting gradients

 

 

x<-b

x =-b

x > -b

x<f

x =f

x>f

x<i

x =i

x >i

f’(x)

-

0

+

+

0

-

-

0

+