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 to show that we may have f'(a) > 0, in contrast to our likely calculus intuition that f'(a) = 0. (iii) Show that ƒ"(a) ≤ 0 by contradiction as follows. If ƒ"(a) > 0, then by continuity f"(x) > 0 for a ≤ x ≤ a + d, with > 0 sufficiently small. Use FTC2 to show that f'(x) > 0 for a≤x≤a+d and thus f is strictly increasing on [a, a + 6]. How does this contradict the maximum occurring at a? (iv) Give examples to show that both f" (a) < 0 and f"(a) = 0 are possible.
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 to show that we may have f'(a) > 0, in contrast to our likely calculus intuition that f'(a) = 0. (iii) Show that ƒ"(a) ≤ 0 by contradiction as follows. If ƒ"(a) > 0, then by continuity f"(x) > 0 for a ≤ x ≤ a + d, with > 0 sufficiently small. Use FTC2 to show that f'(x) > 0 for a≤x≤a+d and thus f is strictly increasing on [a, a + 6]. How does this contradict the maximum occurring at a? (iv) Give examples to show that both f" (a) < 0 and f"(a) = 0 are possible.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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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 to show that we may have f'(a) > 0, in contrast to our likely calculus
intuition that f'(a) = 0.
(iii) Show that ƒ"(a) ≤ 0 by contradiction as follows. If ƒ"(a) > 0, then by continuity
f"(x) > 0 for a ≤ x ≤ a + d, with > 0 sufficiently small. Use FTC2 to show that
f'(x) > 0 for a≤x≤a+d and thus f is strictly increasing on [a, a + 6]. How does this
contradict the maximum occurring at a?
(iv) Give examples to show that both f" (a) < 0 and f"(a) = 0 are possible.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F690bc708-737a-4036-8bde-cd8ee17ec8dd%2F55a9decf-4fad-4182-a74f-230826e44fad%2Fgfw4m66_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 to show that we may have f'(a) > 0, in contrast to our likely calculus
intuition that f'(a) = 0.
(iii) Show that ƒ"(a) ≤ 0 by contradiction as follows. If ƒ"(a) > 0, then by continuity
f"(x) > 0 for a ≤ x ≤ a + d, with > 0 sufficiently small. Use FTC2 to show that
f'(x) > 0 for a≤x≤a+d and thus f is strictly increasing on [a, a + 6]. How does this
contradict the maximum occurring at a?
(iv) Give examples to show that both f" (a) < 0 and f"(a) = 0 are possible.
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