Download
1 / 12
120 likes | 237 Vues
This guide explores the concepts of absolute and local extreme values of functions. It begins by defining absolute maximum and minimum values within a domain, illustrated with examples like the cosine and sine functions. It details the Extreme Value Theorem and local extrema, specifying conditions for when a function attains local maxima or minima at interior points and endpoints. The importance of critical points—where the derivative is zero or undefined—is also discussed. Examples highlight the application of these concepts in determining extreme values.
E N D
Absolute Extreme Values • Let f be a function with domain D. Then f(c) is the • Absolute maximum value on D IFF f(x) ≤ f(c) for all x in D • Absolute minimum value on D IFF f(x) ≥ f(c) for all x in D
Example 1 • On [-π/2, π/2], f(x) = cosx takes on a maximum value of 1 (once) and a minimum value of 0 (twice). The function g(x) = sinx takes on a maximum value of 1 and a minimum value of -1.
Local Extreme Values Let c be an interior point of the domain of the function f. Then f(c) is a • Local maximum value at c IFF f(x) ≤ f(c) for all x in some open interval containing c. • Local minimum value at c IFF f(x) ≥ f(c) for all x in some open interval containing c. A function f has a local maximum or local minimum at an endpoint c if the appropriate inequality holds for all x in some half-open domain interval containing c. *Local extremaare also called relative extrema.
Local Extreme Values Theorem • IF a function f has a local maximum value or a local minimum value at an interior point c of its domain, and if f’ exists at c, then f’(c) = 0
Critical Point • A point in the interior of the domain of a function f at which f’ = 0 or f’ does not exist is a critical point of f.
Example 3 • Find the absolute maximum and minimum values of f(x) = x2/3 on the interval [-2, 3]
Example 4 • Find the extreme values of
Example 5 • Find the extreme values of
Example 6 Find the extreme values of
