4.1 Extreme Values

1. 4.1 Extreme Values

2. Extreme Values • One of the most useful things we can learn about a function is if and where it assumes any max or min values. • Absolute/Global Extreme Values • Let f be a function with domain D. Then f(c) is the • Absolute/Global max on D if and only if f(x) ≤ f(c) for all x in D. • Absolute/Global min on D if and only if f(x) ≥ f(c) for all x in D.

3. Extreme Values y = x2 Domain: (−∞, ∞) Only absolute min

4. Extreme Values Domain: [0, 2] y = x2 Absolute max (at (2, 4)) And Absolute min (at (0, 0))

5. Extreme Values Domain: (0, 2] y = x2 Absolute max (at (2, 4)) only

6. Extreme Values Domain: (0, 2) y = x2 No absolute max or min

7. EXTREME VALUE THEOREM (Big Theorem #2) • If f is continuous on a closed interval [a, b], then f has both a maximum and a minimum value on the interval. max max max max min min min a b a a a b b b min

8. Local/Relative Extreme Values • Let c be an interior point of the domain of the function f. Then f(c) is a • Local/Relative maximum value at c if and only if f(x) ≤ f(c) for all x in some open interval containing c. • Local/Relative minimum value at c if and only if f(x) ≥ f(c) for all x in some open interval containing c. • A function has a local max or local min at an endpoint c if the inequality holds for all x in some half-open domain interval containing c. • An absolute extremum is also a local extremum.

9. Extreme Values Abs. max (also local max) Local min Local max Local min d b a c e Abs. min (also local min)

10. Finding Extreme Values • 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 • A point in the interior of the domain of a function f at which f’ = 0 or f’ does not exist is called a critical point.

11. Finding Extrema • Example: Find the absolute maximum and minimum values of f(x) = x2/3 on the interval [−2, 3].