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? .