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(1) Let f be a function with domain R, with the following four properties: • f is continuous on its entire domain. • lim f(x) = 7. lim f(x) = 10. f(2) = -1. Answer the following questions about f. (a) Prove that f has at least two roots. (Recall: a function g is said to have a root at a point a in its domain if g(a) = 0.) (b) Is it necessarily true that f achieves a minimum value? If you believe this is true, prove it. If you believe it is false, provide a counterexample and explain why it works.

(1) Let f be a function with domain R, with the following four properties: • f is continuous on its entire domain. • lim f(x) = 7. lim f(x) = 10. f(2) = -1. Answer the following questions about f. (a) Prove that f has at least two roots. (Recall: a function g is said to have a root at a point a in its domain if g(a) = 0.) (b) Is it necessarily true that f achieves a minimum value? If you believe this is true, prove it. If you believe it is false, provide a counterexample and explain why it works.

Advanced Engineering Mathematics
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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(1) Let f be a function with domain R, with the following four properties: • f is continuous on its entire domain. • lim f(x) = 7. lim f(x) = 10. f(2) = -1. Answer the following questions about f. (a) Prove that f has at least two roots. (Recall: a function g is said to have a root at a point a in its domain if g(a) = 0.) (b) Is it necessarily true that f achieves a minimum value? If you believe this is true, prove it. If you believe it is false, provide a counterexample and explain why it works.
Transcribed Image Text:(1) Let f be a function with domain R, with the following four properties: • f is continuous on its entire domain. • lim f(x) = 7. lim f(x) = 10. f(2) = -1. Answer the following questions about f. (a) Prove that f has at least two roots. (Recall: a function g is said to have a root at a point a in its domain if g(a) = 0.) (b) Is it necessarily true that f achieves a minimum value? If you believe this is true, prove it. If you believe it is false, provide a counterexample and explain why it works.
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