How do you use the second derivative test how do you find the local maxima and minima of #f(x) = 12 + 2x^2 - 4x^4#?

1 Answer
Bill K.
May 19, 2015

The function #f(x)=12+2x^2-4x^4# has derivative #f'(x)=4x-16x^3# and second derivative #f''(x)=4-48x^2#.

The critical points occur where #f'(x)=4x(1-4x^2)=0#, which are #x=0# and #x=\pm 1/2#.

Since #f''(\pm 1/2)=4-48\cdot 1/4=4-12=-8<0#, the second derivative test says the critical points at #x=\pm 1/2# are local maxima (the graph of #f# is concave down near #x=\pm 1/2#).

Since #f''(0)=4>0#, the second derivative test says the critical point at #x=0# is a local minimum (the graph of #f# is concave up near #x=0#).

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