Question

How do you find all critical point and determine the min, max and inflection given `f(x)=x^4`?

Answer 1

`f(x)=x^4`

Domain of `f` is `(-oo,oo)`.

A critical number for `f` is a number `c` in the domain of `f` at which `f'(c)` does not exist or `f'(c)=0`

`f'(x) = 4x^3` is defined for all `x` and is `0` at `x=0`.

The only critical number for `f` is `0`.

(If your treatment of calculus says that a critical point is a point on the graph, then the critical point is `(0,0)`)

Because `f'(x)` is negative for `x < 0` and positive for `x > 0`, we know that `f(0) = 0` is a local minimum.

`f''(x) = 12x^2` which is always positive.

Since the sign of `f''` never changes, the concavity of `f` never changes, so there are no inflection points.