How do you graph f(x)=(x^2-4)^2 using the information given by the first derivative?

1 Answer
Feb 17, 2017

f(x) has local minima at x=+-2 and a local maximum at x=0

Explanation:

f(x) = (x^2-4)^2

f'(x) = 2(x^2-4)*2x [Chain rule]
=4x^3-16x

For extrema f'(x) =0

4x^3-16x=0

4x(x^2-4)=0

4x(x+2)(x-2)=0

x=0 or+-2

f''(x) = 12x^2-16

f''(0) <0 -> f(0) is a local maximun

f''(+-2) >0 -> f(+-2) are local minimum

These results can be seen on the graph of f(x) below:

graph{(x^2-4)^2 [-23.44, 22.18, -0.34, 22.45]}