3.2: Extrema and the First Derivative Test

Definition of Relative Extrema Let f be a function defined at c. f(c) is a relative maximum of f if there exists an interval (a, b) containing c such that for all x in (a, b). f(c) is a relative minimum of f if there exists an interval (a, b) containing c such that for all x in (a, b).

Where? If f has a relative minimum or relative maximum when x = c, then c is a critical number of f.

1st Derivative Test If c is a critical number of f, then f can be classified as a relative minimum, a relative maximum, or neither: If changes from negative to positive at x = c, then f(c) is a relative minimum. If changes from positive to negative at x = c, then f(c) is a relative maximum. If does not change sign at x = c, then f(c) is not a relative extrema.

Find all relative extrema:

Find all relative extrema:

Find all relative extrema:

Absolute Extrema Let f be defined on an interval I containing c. is an absolute minimum of f on I if for all x in I. is an absolute maximum of f on I if for all x in I.

Extreme Value Theorem If f is continuous on [a, b], then f has both a minimum value and a maximum value on [a, b]. Absolute max. and absolute min. values can occur at either critical values or endpoints.

To find extrema on [a, b] Find critical numbers of f. Evaluate f at the critical numbers in the interval. Evaluate f at the endpoints, a and b. The smallest is the minimum, the largest is the maximum!

Example Find the minimum and maximum values of f(x) = x2 – 8x + 10 on the interval [0, 7].

Example The concentration C (in milligrams per milliliter) of a chemical in the bloodstream t hours after injection into muscle tissue can be modeled by Determine the time when the concentration has reached a maximum.