# Second Derivative Test

Instead of using First Derivative Test we can use another test.

**Second Derivative Test.**

- if `f'(c)=0` and `f''(c)<0` then there is maximum at point `c`.
- if `f'(c)=0` and `f''(c)>0` then there is minimum at point `c`.
- if `f'(c)=0` and `f''(c)=0` then we can't say anything about point `c`.

This test is used not so often as first derivative test because of two reasons:

- We can't apply it to stationary points for which first derivative doesn't exist (because in this case second derivative also doesn't exist).
- It is inconclusive when second derivative equals 0.

**Example.** Find extrema of the function `y=x^4-4x^3`.

We first find first two derivatives.

`f'(x)=4x^3-12x^2=4x^2(x-3)`.

So, critical numbers (points where `f'(x)=0`) are `x=0,x=3`.

`f''(x)=(f'(x))'=(4x^3-12x^2)'=12x^2-24x=12x(x-2)`.

Since `f''(3)=12*3*(3-2)=36>0` then according to second derivative test `x=3` is minimum.

Since `f''(0)=12*0(0-2)=0` then we can't say anything about `x=0` using second derivative test. Using first derivative test we conclude that `x=0` is neither minimum nor maximum.