MAXIMUM AND MINIMUM VALUES THE EXTREME VALUE THEOREM
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MAXIMUM AND MINIMUM VALUES THE EXTREME VALUE THEOREM 1 2 f(x) is continuous on attains an absolute maximum f(c) and minimum f(d) value Closed interval [a, b] How to find absolute Max and Min 1. Find all critical points ( ƒ’= 0 , or ƒ’ undefined ) 2. Evaluate ƒ at all criticals and endpoints a, b 3. Take the largest value and the smallest EXAMPLE: Find: Absolute max and min EXAMPLE: Find: Absolute max and min (not closed interval) How to find absolute Max and Min
MAXIMUM AND MINIMUM VALUES How to find absolute Max and Min not closed interval (a, b) 1. Find all critical points ( ƒ’= 0 , or ƒ’ undefined ) 2. Evaluate ƒ at all criticals and 3. Take the largest value and the smallest (exclude it if at a or b) EXAMPLE: Find: Absolute max and min smallest (global min) Largest (global max)
MAXIMUM AND MINIMUM VALUES How to find absolute Max and Min not closed interval [a, b) 1. Find all critical points ( ƒ’= 0 , or ƒ’ undefined ) 2. Evaluate ƒ at all criticals and endpoint a, 3. Take the largest value and the smallest (exclude it if at b) F 083 smallest (global min) Largest (no global max)
MAXIMUM AND MINIMUM VALUES How to find absolute Max and Min not closed interval (a, b) 1. Find all critical points ( ƒ’= 0 , or ƒ’ undefined ) 2. Evaluate ƒ at all criticals and 3. Take the largest value and the smallest (exclude it if at a or b) EXAMPLE: Find: Absolute max and min Largest (no global max) smallest (global min) Largest (no global max)
MAXIMUM AND MINIMUM VALUES How to find absolute Max and Min not closed interval 1. Find all critical points ( ƒ’= 0 , or ƒ’ undefined ) 2. Evaluate ƒ at all criticals and 3. Take the largest value and the smallest (exclude it if at a or b) EXAMPLE: Find: Absolute max and min
MAXIMUM AND MINIMUM VALUES EXAMPLE: Find: Absolute max and min Remarks: Polynomial with odd degree no absolute max, no absoulte min Polynomial with even degree absolute max only or absoulte min only