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

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Answer 1 · 2017-02-17T10:51:04

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

#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]}