4.1 Extreme Values on Functions

Vocab/Formulas Extreme Values: Absolute/Global Extrema: Absolute Max: f(c) iff f(x)≤f(c) for all x Absolute Min: f(c) iff f(x)≥f(c) for all x Local/Relative Extrema: Local Max: f(c) iff f(x)≤f(c) for all x in some open interval containing c Local Min: f(c) iff f(x)≥f(c) for all x in some open interval containing c

Experiment: Where do max’s and min’s happen?

Local Extreme Values Theorem 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. 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

EXTREME VALUES OCCUR ONLY AT CRITICAL POINTS AND ENDPOINTS Vocab/Formulas Critical Point: A point on the interior of a function f at which f ’ = 0 or f ’ DNE EXTREME VALUES OCCUR ONLY AT CRITICAL POINTS AND ENDPOINTS

Example 1: Find the extreme values for the given graphs.

Example 2: Find the extreme values and where they occur for the function on the given interval.   Graphically Analytically

Example 3: Find the extreme values and where they occur for the function on the given interval.   Graphically Analytically

Example 4: Find the extreme values and where they occur for the function.   Graphically Analytically

Assignment Pg. 193 1-13, 17