Ⓒ✓H045. Suppose that ƒ : R → R is continuous. and f(x) > 0.. Prove that there is an in-terval I = (xo-=,xo+=), where n is natural number, such that f(x) > 0 for all x in I.Q Search2 CMeQI

Given that f: R→R is a continuous function and f(x0)>0

Answered: 5. Suppose that f: R→ R is continuous.…

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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as(3) = { kx + 5 for x 2 - Find the value of k for which f(x) is continuous at x=2. O A) -2 O B) -1 O C) 0 O D) 1 O E) 7
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4. Let f: [-1, 1] → R be a continuous function. Prove the following statements: (a) If there is c € [-1,1] such that f(c)f(−c) < 0, then there is de R such that f(d) = 0. (b) If ƒ([-1, 1]) = (-1, 1), then f is not continuous.
2.) Decide whether the statement is either true or false: Circle T or F to indicate your answer. Assume that f is a function. No work needs to be shown. a.) T or F If f (x)>0 on the interval (-x,3) and f'(x)0 on the intervals (-,3) and (3,0), then f must have a local minimum at the x-value 3. c.) T or F If f is continuous on (-0,0), and f"(x) >0 on the interval (-0,3) and f"(x)<0 on (3,0), then f must have an inflection point at the x-value 3. d.) T or F The limit lim -is of indeterminate form. In(x) x-7 -is of indeterminate form. e.) T or F The limit lim- x-l" x-1
Suppose f is a continuous function defined on I = [a, b] such that f'(r) exists for all z € I (but f' need not itself be continuous). Suppose further that f'(a) < f'(b). Prove for any y with f'(a) < y < f'(b) that there is some ce I such that f'(c) = y.
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5. Suppose that f: R→ R is continuous. and f(ro) > 0.. Prove that there is an in- terval I = (xoxo+), where n is natural number, such that f(x) > 0 for all x in I.