How do you use the second derivative test how do you find the local maxima and minima of f(x) = 12 + 2x^2 - 4x^4f(x)=12+2x24x4?

1 Answer
May 19, 2015

The function f(x)=12+2x^2-4x^4f(x)=12+2x24x4 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).