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5. Consider a, b in R where a < b. Show there exist infinitely differentiable functions fa, gb, ha,b, and hon R with the following properties. (a) fa(x) = 0 for x 0 for x > a. (b) g(x) =0 for x ≥b and g(x) > 0 for x 0 for x € (a, b) and hab(x) = 0 for x (a, b). (d) h(x) = 0 for x ≤a and h(x) = 1 for x> b.

5. Consider a, b in R where a < b. Show there exist infinitely differentiable functions fa, gb, ha,b, and hon R with the following properties. (a) fa(x) = 0 for x 0 for x > a. (b) g(x) =0 for x ≥b and g(x) > 0 for x 0 for x € (a, b) and hab(x) = 0 for x (a, b). (d) h(x) = 0 for x ≤a and h(x) = 1 for x> b.

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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5. Consider a, b in R where a < b. Show there exist infinitely differentiable functions fa, gb, ha,b, and hon R with the following properties. (a) fa (x) = 0 for x ≤ a and fa(x) > 0 for x > a. (b) g(x) = 0 for x ≥b and g(x) > 0 for x < b. (c) hab(x) > 0 for x = (a, b) and hab(x) = 0 for x (a, b). (d) h(x)=0 for x≤ a and h(r) = 1 for x ≥b.
Transcribed Image Text:5. Consider a, b in R where a < b. Show there exist infinitely differentiable functions fa, gb, ha,b, and hon R with the following properties. (a) fa (x) = 0 for x ≤ a and fa(x) > 0 for x > a. (b) g(x) = 0 for x ≥b and g(x) > 0 for x < b. (c) hab(x) > 0 for x = (a, b) and hab(x) = 0 for x (a, b). (d) h(x)=0 for x≤ a and h(r) = 1 for x ≥b.
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