calculus_solutions_4_1 - f x = c f ( c ) f f c f ( c ) x c...

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Unformatted text preview: f x = c f ( c ) f f c f ( c ) x c b d b e d s a c r t e t c e s b c d r a f (4)=4 f (7)=0 f (4)=4 f (6)=3 f (2)=1 f (5)=2 f (8)=5 f (2)=0 f (1)=2 f (4)=4 f (6)=3 f (2)=0 f (5)=2 f (7)=1 1. A function has an absolute minimum at if is the smallest function value on the entire domain of , whereas has a local minimum at if is the smallest function value when is near . 2. (a) The Extreme Value Theorem (b) See the Closed Interval Method. 3. Absolute maximum at ; absolute minimum at ; local maxima at and ; local minima at and ; neither a maximum nor a minimum at , , , and . 4. Absolute maximum at ; absolute minimum at ; local maxima at , , and ; local minima at , , , and ; neither a maximum nor a minimum at . 5. Absolute maximum value is ; absolute minimum value is ; local maximum values are and ; local minimum values are and . 6. Absolute maximum value is ; absolute minimum value is ; local maximum values are , , and ; local minimum values are , , and . 7. Absolute minimum at 2, absolute maximum at 3, local minimum at 4 8. Absolute minimum at 1, absolute maximum at 5, local maximum at 2, local minimum at 4 9. Absolute maximum at 5, absolute minimum at 2, local maximum at 3, local minima at 2 and 4 1 Stewart Calculus ET 5e 0534393217;4. Applications of Differentiation; 4.1 Maximum and Minimum Values f 10. has no local maximum or minimum, but 2 and 4 are critical numbers 11. (a) (b) (c) 12. (a) Note that a local maximum cannot occur at an endpoint. 2 Stewart Calculus ET 5e 0534393217;4. Applications of Differentiation; 4.1 Maximum and Minimum Values f f (b) Note: By the Extreme Value Theorem, must not be continuous. 13. (a) Note: By the Extreme Value Theorem, must not be continuous; because if it were, it would attain an absolute minimum. (b) 14. (a) 3 Stewart Calculus ET 5e 0534393217;4. Applications of Differentiation; 4.1 Maximum and Minimum Values f ( x )=8 3 x x 1 f (1)=5 f ( x )=3 2 x x 5 f (5)= 7 f ( x )= x 2 0< x <2 (b) 15. , . Absolute maximum ; no local maximum. No absolute or local minimum. 16. , . Absolute minimum ; no local minimum. No absolute or local maximum. 17. , . No absolute or local maximum or minimum value. 4 Stewart Calculus ET 5e 0534393217;4. Applications of Differentiation; 4.1 Maximum and Minimum Values f ( x )= x 2 0< x 2 f (2)=4 f ( x )= x 2 x <2 f (0)=0 f ( x )= x 2 x 2 f (2)=4 f (0)=0 f ( x )= x 2 3 x 2 f ( 3)=9 f (0)=0 18. , . Absolute maximum ; no local maximum. No absolute or local minimum. 19. , . Absolute minimum ; no local minimum. No absolute or local maximum. 20. , . Absolute maximum . Absolute minimum . No local maximum or minimum. 21. , . Absolute maximum . No local maximum. Absolute and local minimum ....
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This note was uploaded on 11/04/2010 for the course MATH 1110 taught by Professor Martin,c. during the Spring '06 term at Cornell University (Engineering School).

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calculus_solutions_4_1 - f x = c f ( c ) f f c f ( c ) x c...

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