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

1 Answer
Oct 26, 2016

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.