ISM_T11_C04_A - CHAPTER 4 APPLICATIONS OF DERIVATIVES 4.1...

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CHAPTER 4 APPLICATIONS OF DERIVATIVES 4.1 EXTREME VALUES OF FUNCTIONS 1. An absolute minimum at x c , an absolute maximum at x b. Theorem 1 guarantees the existence of such œœ # extreme values because h is continuous on [a b]. ß 2. An absolute minimum at x b, an absolute maximum at x c. Theorem 1 guarantees the existence of such extreme values because f is continuous on [a b]. ß 3. No absolute minimum. An absolute maximum at x c. Since the function's domain is an open interval, the œ function does not satisfy the hypotheses of Theorem 1 and need not have absolute extreme values. 4. No absolute extrema. The function is neither continuous nor defined on a closed interval, so it need not fulfill the conclusions of Theorem 1. 5. An absolute minimum at x a and an absolute maximum at x c. Note that y g(x) is not continuous but œ still has extrema. When the hypothesis of Theorem 1 is satisfied then extrema are guaranteed, but when the hypothesis is not satisfied, absolute extrema may or may not occur. 6. Absolute minimum at x c and an absolute maximum at x a. Note that y g(x) is not continuous but still œ has absolute extrema. When the hypothesis of Theorem 1 is satisfied then extrema are guaranteed, but when the hypothesis is not satisfied, absolute extrema may or may not occur. 7. Local minimum at , local maximum at ab a b ±"ß ! "ß ! 8. Minima at and , maximum at a b ±# ß! # ! ß# 9. Maximum at . Note that there is no minimum since the endpoint is excluded from the graph. !ß & #ß ! 10. Local maximum at , local minimum at , maximum at , minimum at a b a b ±$ # " ! ß±" 11. Graph (c), since this the only graph that has positive slope at c. 12. Graph (b), since this is the only graph that represents a differentiable function at a and b and has negative slope at c. 13. Graph (d), since this is the only graph representing a funtion that is differentiable at b but not at a. 14. Graph (a), since this is the only graph that represents a function that is not differentiable at a or b.
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198 Chapter 4 Applications of Derivatives 15. f(x) x 5 f (x) no critical points; œ± Ê œ Ê 22 33 w f( 2) , f(3) 3 the absolute maximum ±œ ± œ
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This note was uploaded on 09/20/2010 for the course MATHEMATIC 09991051 taught by Professor Dr.maenshadeed during the Fall '10 term at Norwegian Univ. of Science & Technology.

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ISM_T11_C04_A - CHAPTER 4 APPLICATIONS OF DERIVATIVES 4.1...

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