Calculus BC: Applications of the Derivative

Terms

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 .

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