4.1 Maximum and Minimum Values

Maximum Values Local Maximum Absolute Maximum |c2|c2 |c1|c1 I

Minimum Values Local Minimum Absolute Minimum I c2|c2| |c1|c1 I Collectively, maximum and minimum values are called extreme values.

Definitions A function f has an absolute (global) maximum at c if f (c) ≥ f (x) for all x in the domain. A function f has an absolute (global) minimum at c if f (c) ≤ f (x) for all x in the domain. The maximum and minimum values are called extreme values. A function f has a local (relative) maximum at c if f (c) ≥ f (x) where x is in a small open interval about c. A function f has a local (relative) minimum at c if f (c) ≤ f (x) where x is in a small open interval about c.

The Extreme Value Theorem If f is continuous on a closed interval [a, b], then f attains an absolute maximum value f (c) and an absolute minimum f (d) at some numbers c and d in [a, b]. ] ● ● [

Fermat’s Theorem If f has a local maximum or minimum at c, and if f '(c) exists, then f ' (c) = 0. Question If a (local) maximum or minimum occur at c, then what is the value of f '(c)?

Critical Number or Value A critical number of a function f is a number c in the domain of f such that either f '(c) = 0 or f '(c) does not exist ( f (x) is not differentiable) Fact An absolute extremum occurs at two places:  Critical points  End points

Finding Absolute Extrema on a Closed Interval [a,b] 1.Find the critical numbers of f on (a, b). 2.Compute the value of f at each of the critical numbers on (a,b) the endpoints a and b. 3.The largest of these values is the absolute maximum. The smallest is the absolute minimum.

Examples Locate the absolute extrema of the function on the closed interval. f(x) = x 3 – 12x on [-3,4] g(x) = 4x / (x 2 +1) on [0,3] h(t) = 2 sec(t) - tan(t) on [0,π/4]