Q2 MULTIPLE CHOICE One answer only If f: (0, ∞) → R is differentiable and f' is bounded below by some constant a > 0, then ƒ is not bounded. a. True, because f(x) - f(1) ≥ a(x - 1) whenever x > 1 (by MVT), so limx→∞ f (x) = +∞. b. False, if f' is bounded then ƒ is bounded by MVT. c. True, because f(x) = f(1) ≥ a(x - 1) whenever x > 1 (by fundamental theorem of calculus), so limx→∞ f (x) = +∞. d. False, here is a counter-example: f(x) = 1- e-ª.
Q2 MULTIPLE CHOICE One answer only If f: (0, ∞) → R is differentiable and f' is bounded below by some constant a > 0, then ƒ is not bounded. a. True, because f(x) - f(1) ≥ a(x - 1) whenever x > 1 (by MVT), so limx→∞ f (x) = +∞. b. False, if f' is bounded then ƒ is bounded by MVT. c. True, because f(x) = f(1) ≥ a(x - 1) whenever x > 1 (by fundamental theorem of calculus), so limx→∞ f (x) = +∞. d. False, here is a counter-example: f(x) = 1- e-ª.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Q2MULTIPLE CHOICE One answer only
If ƒ : (0, ∞) → R is differentiable and f' is bounded below by some constant a > 0, then ƒ is not bounded.
a. True, because ƒ(x) − ƒ(1) ≥ a(x − 1) whenever x > 1 (by MVT), so limx→∞ f (x) = +∞.
b. False, if f' is bounded then f is bounded by MVT.
c. True, because f(x) − ƒ(1) ≥ a(x − 1) whenever x > 1 (by fundamental theorem of calculus), so
limx→∞ f (x) = +∞.
d. False, here is a counter-example: f(x) = 1−e¯ª.
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