AP Calculus BC 4.1 Extreme Values of Functions Objective: able to determine the local or global extreme values of a function.

Absolute (Global) Extreme Values

Let f be a function with domain Dfc . Then ( ) is the (a) absolute maximum value on D if and only if fxfc ( )≤ ( ) for all xD in .

(b) absolute minimum value on D if and only if fxfc ( )≥ ( ) for all xD in .

1. Function Rule Domain D Absolute Extrema on D

2 a. y = – x + 1 −∞, ∞

b. y = – x 2 + 1 [-2, 0]

c. y = – x 2 + 1 [-2, 0)

d. y = – x 2 + 1 (-2, 0)

Extreme Value Theorem If f is continuous on a closed interval [ a, b], then f has both a maximum value and a minimum value on the interval. The maximum and minimum values may occur at interior points or endpoints.

Local (Relative) Extreme Values Let c be an interior point of the domain of the function f . Then f ( c ) is a

(a) local maximum value at c if and only if fxfc ( )≤ ( ) for all x in some

open interval containing c . (b) local minimum value at c if and only if fxfc ( )≥ ( ) 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-op en domain

interval containing c .

Local Extreme Value 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 cfc , then '( )= 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 . Extreme values occur only at critical points and endpoints. However, not every critical point or endpoint signals the presence of an extreme value.

2. Inspect f (x) = x 5 and f -1(x).

3. Find the extreme values of and where they occur. = √

4. Identify the critical points and determine the extreme values for the function − 2 + 4, ≤ 2 . = − 6 + 12, > 2

5. Identify the critical points and determine the extreme values for = √4 −

Rate yourself on how well you understood this lesson. I understand I don’t get I sort of I understand most of it but I I got it! it at all get it it pretty well need more practice 1 2 3 4 5 What do you still need to work on?