
Critical Point
A number x in the domain of a function f such that f'(x) = 0.

Global Maximum
A value f (x_{0}) that is greater than or equal to any other value f (x) that f takes on over its domain.

Global Minimum
A value f (x_{0}) that is less than or equal to any other value f (x) that f takes on over its domain.

Inflection Point
A number x in the domain of a function f such that f’’(x) = 0.

Local Maximum
A value f (x_{0}) that is greater than or equal to any other value f (x) for x in some interval about x_{0}.

Local Minimum
A value f (x_{0}) that is less than or equal to any other value f (x) for x in some interval about x_{0}.

First Derivative Test
A critical point x_{0} of a function f is a local maximum if the first derivative f' changes sign from positive to negative at x_{0}. Correspondingly, x_{0} is a local minimum is f' changes sign from negative to positive there.

Second Derivative Test
A critical point x_{0} of a function f is a local maximum if the second derivative f''(x_{0}) is negative. It is a local minimum if f''(x_{0}) is positive. (It is also possible that f''(x_{0}) = 0, in which case the critical point is also an inflection point.)

Concave Up
A function f (x) is concave up at x_{0} if f''(x_{0}) > 0.

Concave Down
A function f (x) is concave down at x_{0} if f''(x_{0}) < 0.