-
Critical Point
A number x in the domain of a function f such that f'(x) = 0.
-
Global Maximum
A value f (x0) that is greater than or equal to any other value f (x) that f takes on over its domain.
-
Global Minimum
A value f (x0) 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 (x0) that is greater than or equal to any other value f (x) for x in some interval about x0.
-
Local Minimum
A value f (x0) that is less than or equal to any other value f (x) for x in some interval about x0.
-
First Derivative Test
A critical point x0 of a function f is a local maximum if the first derivative f' changes sign from positive to negative at x0. Correspondingly, x0 is a local minimum is f' changes sign from negative to positive there.
-
Second Derivative Test
A critical point x0 of a function f is a local maximum if the second derivative f''(x0) is negative. It is a local minimum if f''(x0) is positive. (It is also possible that f''(x0) = 0, in which case the critical point is also an inflection point.)
-
Concave Up
A function f (x) is concave up at x0 if f''(x0) > 0.
-
Concave Down
A function f (x) is concave down at x0 if f''(x0) < 0.