Question

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`?

Answer 1

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`).