Ⓒ✓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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Give an example of a function which is continuous on [0,1] except at x = 1/2 and for which the intermediate value theorem does not hold
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
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Suppose that the function f: R → R has the property that -x2 <ƒ(x)
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15. Let f. g : R → R be continuous at a point c, and let h(x) := sup (f (x). 8(x)} for x € R. Show that h(x) = (f(x) + g(x)} + }If(x) – g(x)| for all x € R. Use this to show that h is continuous at c.
Assume f(a, b) → R is differentiable at some point c = (a, b). Prove that if f'(c) 0, then there exists some > 0, such that f(x) + f(c) for all x € (c-d, c+d).

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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.