25.4 Problem. In calculus, we usually apply derivative tests for extreme values occurring at interior points of intervals, so here is a chance to think about what happens at the endpoints. Let f = C²([a, b]) and suppose that f(a) = max f(x). a≤x≤b (i) Use the definition of the derivative to prove that f'(a) ≤ 0. (ii) Give an example (start by drawing a picture) to show that we may have f'(a) < 0, in contrast to our likely calculus intuition that f'(a) = 0. (iii) Give examples (start, again, by drawing pictures) to show that any of the possibilities f" (a) > 0, f" (a) = 0, or f" (a) < 0 are possible. When drawing, remember that f'(a) <0.
25.4 Problem. In calculus, we usually apply derivative tests for extreme values occurring at interior points of intervals, so here is a chance to think about what happens at the endpoints. Let f = C²([a, b]) and suppose that f(a) = max f(x). a≤x≤b (i) Use the definition of the derivative to prove that f'(a) ≤ 0. (ii) Give an example (start by drawing a picture) to show that we may have f'(a) < 0, in contrast to our likely calculus intuition that f'(a) = 0. (iii) Give examples (start, again, by drawing pictures) to show that any of the possibilities f" (a) > 0, f" (a) = 0, or f" (a) < 0 are possible. When drawing, remember that f'(a) <0.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.6: The Inverse Trigonometric Functions
Problem 91E
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![25.4 Problem. In calculus, we usually apply derivative tests for extreme values occurring
at interior points of intervals, so here is a chance to think about what happens at the
endpoints. Let f = C²([a, b]) and suppose that
f(a) = max f(x).
a≤x≤b
(i) Use the definition of the derivative to prove that f'(a) ≤ 0.
(ii) Give an example (start by drawing a picture) to show that we may have f'(a) < 0, in
contrast to our likely calculus intuition that f'(a) = 0.
(iii) Give examples (start, again, by drawing pictures) to show that any of the possibilities
f" (a) > 0, f" (a) = 0, or f" (a) < 0 are possible. When drawing, remember that f'(a) <0.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F690bc708-737a-4036-8bde-cd8ee17ec8dd%2Fd0b3f291-7e36-46e7-9d42-3797a71181cb%2Fu14lkhh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:25.4 Problem. In calculus, we usually apply derivative tests for extreme values occurring
at interior points of intervals, so here is a chance to think about what happens at the
endpoints. Let f = C²([a, b]) and suppose that
f(a) = max f(x).
a≤x≤b
(i) Use the definition of the derivative to prove that f'(a) ≤ 0.
(ii) Give an example (start by drawing a picture) to show that we may have f'(a) < 0, in
contrast to our likely calculus intuition that f'(a) = 0.
(iii) Give examples (start, again, by drawing pictures) to show that any of the possibilities
f" (a) > 0, f" (a) = 0, or f" (a) < 0 are possible. When drawing, remember that f'(a) <0.
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