Unit 4: Applications of Derivatives Sec 1: Extrema and Critical Points

I. Definition of extrema Also called Extreme Values Let f be a function defined on an interval, I, containing c. 1. f(c) is the maximum of f on I if f(c) ≥ f(x) for all x in I. 2. f(c) is the minimum of f on I if f(c) ≤ f(x) for

Extreme value theorem If f is continuous on a closed interval [a, b], then f has both a maximum and a minimum on the interval. Ex. Graph y = x² + 1 on the intervals [-1, 2] and (-1, 2) to demonstrate.

Relative Extrema/Critical numbers What is the difference between a absolute and a relative maximums and minimums?

II. Definition of critical numbers Let f be defined at c. If f’(c) = 0 or if f is not differentiable at c, then c is a critical number of f. Critical numbers theorem If f has a relative max or min at x = c, then c is a critical number of f.

Guidelines for finding extrema on a closed interval Find the critical #’s of f on (a, b) by solving f’(x) = 0 and locating where the Derivative Does Not Exist. Evaluate f at each critical # in (a, b) by substituting the critical # in f(x). Evaluate f at each endpoint of [a, b]. Determine which critical value/endpoint is the maximum and minimum. The least y-value is the minimum The greatest y-value is the maximum.

Ex 1: find the extrema of on the interval [-1, 2].

homework Pg 165 #17-31 odds