Determine if the following statements are true or false. a. If a function f has an absolute maximum on the interval [−1,3], then it also has an absolute minimum on the same interval. True False b. If f'(−4)=0 and f'(x)<0 for x<−4, then f is changing from decreasing to increasing at x=−4. False True c. If a function f is continuous on the interval [0,∞), then it has either an absolute maximum or absolute minimum on the interval. FalseTrue d. On the interval [−1,1], the function f(x)=1−x2/3 satisfies the conditions of Rolle's theorem. False True e. If f is an everywhere continuous and differentiable function where f'(1)=0 and f''(1)=0, then the point corresponding to x=1is neither a relative maximum nor relative minimum of f. False True

Determine if the following statements are true or false. a. If a function f has an absolute maximum on the interval [−1,3], then it also has an absolute minimum on the same interval. True False b. If f'(−4)=0 and f'(x)<0 for x<−4, then f is changing from decreasing to increasing at x=−4. False True c. If a function f is continuous on the interval [0,∞), then it has either an absolute maximum or absolute minimum on the interval. FalseTrue d. On the interval [−1,1], the function f(x)=1−x2/3 satisfies the conditions of Rolle's theorem. False True e. If f is an everywhere continuous and differentiable function where f'(1)=0 and f''(1)=0, then the point corresponding to x=1is neither a relative maximum nor relative minimum of f. False True

Algebra: Structure And Method, Book 1

(REV)00th Edition

ISBN:9780395977224

Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Chapter2: Working With Real Numbers

Section2.9: Dividing Real Numbers

Problem 3MRE

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Question

Determine if the following statements are true or false.

a. If a function f has an absolute maximum on the interval

[−1,3], then it also has an absolute minimum on the same interval.

True

False

b. If f'(−4)=0 and f'(x)<0 for x<−4, then f is changing from decreasing to increasing at x=−4.

False

True

c. If a function f is continuous on the interval [0,∞), then it has either an absolute maximum or absolute minimum on the interval.

FalseTrue

d. On the interval [−1,1], the function f(x)=1−x^2/3 satisfies the conditions of Rolle's theorem.

False

True

e. If f is an everywhere continuous and differentiable function where f'(1)=0 and f''(1)=0, then the point corresponding to x=1is neither a relative maximum nor relative minimum of f.

False

True

Formula Formula A function f(x) attains a local maximum at x=a , if there exists a neighborhood (a−δ,a+δ) of a such that, f(x)<f(a), ∀ x∈(a−δ,a+δ),x≠a f(x)−f(a)<0, ∀ x∈(a−δ,a+δ),x≠a In such case, f(a) attains a local maximum value f(x) at x=a .

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