05Minimum and Maximum Problems

# 05Minimum and Maximum Problems - MINIMUM AND MAXIMUM...

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M INIMUM AND M AXIMUM P ROBLEMS Abs. min but no no absolute extrema both an abs. max and abs max on ) , ( -∞ on ) , ( -∞ abs min on ) , ( -∞ a b a b Both abs. max and none on [ ] b a , both on [ ] b a , min on [ ] b a , a b ) ( x f is not continuous on [ ] b a , so no abs. To find the absolute extrema of a continuous function f on a finite closed interval [ ] b a , . 1. Find the critical numbers of f in ( 29 b a , . 2. Evaluate f at all the critical numbers and at the endpoints a and b. 3. The largest of these values is the absolute max, the smallest is the absolute min. Applied Problems: Step 1: Draw and label a picture. Step 2: Find a formula for the quantity to be maximized or minimized. Step 3: Express the quantity as a function of 1 variable. Step 4: Find the restrictions. Step 5: Find the max or min. Extreme-Value Theorem : If a function f is continuous on a finite closed interval [ ] b a , , then f has both an absolute maximum and an absolute minimum on [ ] b a , .

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