Question

Let f'' (x)= 4x^3 - 2x and let f(x) has critical numbers -1, 0, and 1, how do you use the second derivative test to determine which critical numbers, if any give a relative maximum?

Answer 1

For each critical number `c`, find `f''(c)`.
If `f''(c) >0`, then `f(c)` is a relative minimum,
if `f''(c) < 0`, then `f(c)` is a relative maximum,
`f''(c) = 0`, then the test fails (We would use the first derivative test if we knew `f'(x)`.)

For example: `f''(-1) = 4(-1)^3-2(-1) =-4 +2 < 0` so `f(-1)` is a relative maximum.

( Note: You can say more about the critical number `0` in this problem after you know how to "work backwards" from `f''(x)` to `f'(x)`.)