Let f be a function defined for all x, and continuous at all values of x except at -2 and 0. bx² + x x²-b Moreover, suppose that the following is for all x > 10 we have that b ≤ f(x) ≤. lim_f(x) = 3, x 2 lim f(x) = f(-2) = 1, lim f(x) = f(0) = -b, x 2+ x-0 lim f(- lim f(x). x->0+ (a) Does there always exist a value c on the interval (-3,0) such that f(c) = 0? Justify (b) Does there always exist a value c on the interval (-1,1) such that f(c) = 0? Justify (c) Evaluate the following limit:
Let f be a function defined for all x, and continuous at all values of x except at -2 and 0. bx² + x x²-b Moreover, suppose that the following is for all x > 10 we have that b ≤ f(x) ≤. lim_f(x) = 3, x 2 lim f(x) = f(-2) = 1, lim f(x) = f(0) = -b, x 2+ x-0 lim f(- lim f(x). x->0+ (a) Does there always exist a value c on the interval (-3,0) such that f(c) = 0? Justify (b) Does there always exist a value c on the interval (-1,1) such that f(c) = 0? Justify (c) Evaluate the following limit:
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:Let f be a function defined for all x, and continuous at all values of x except at -2 and 0. Suppose that
bx² + x
for all x > 10 we have that b ≤ f(x) ≤2. Moreover, suppose that the following is known:
lim_f(x) = 3,
x 2
lim f(x) = f(-2) = 1, lim f(x) = f(0) = -b,
x 2+
x-0
lim_ f(x) = 2.
x-0+
(a) Does there always exist a value c on the interval (-3,0) such that f(c) = 0? Justify
(b) Does there always exist a value c on the interval (-1,1) such that f(c) = 0? Justify
(c) Evaluate the following limit:
lim f(x).
Justify
In (a) and (b) we want to know if there always exists such a value c. Finding a specific example
where such a c exists does not show that something always happens. The only thing you are allowed to
assume about f(x) is the information given in the prompt.
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