5. Assumef: [0, 1] → [0, 1] is continuous with f(0) = f(1). Prove thereare x, y = [0, 1] with |x - y = and f(x) = f(y)

We have to assume f: [0, 1] → [0, 1] is continuous with f(0) = f(1). and prove that there are x, y ∈…

Answered: 5. Assume f: [0, 1] → [0, 1] is…

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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7. Modify B(z) to construct a C function C(z) which satisfies a. C(z) = 0 if S 0.. b. C(z) = 1 if > 1. c. C'(z) > 0 if0< z < 1.
3. Define f(x) so that 16) = { exp (교1), |피 1 f(r) = 0, (a) Explain why the function f(x) must be continuous everywhere except perhaps at x = +1. Hint: How is f(x) constructed from known continuous functions? (b) State the definition for f(x) to be continuous at c. Assume that c is not an endpoint. (c) Prove that f(x) is in fact continuous at both –1 and 1. (d) State the limit definition for f(x) to be differentiable at c. Assume that c is not an endpoint. (e) Explain why the function f(x) must be differentiable everywhere except perhaps at r = ±1. Hint: How is f(x) constructed from known differentiable functions? (f) Prove that f(x) is in fact differentiable at both -1 and 1.
2. Sketch the graph of one function g that is continuous on [-4, 4] and has all of the following properties: (a) g(-4) = 1 = g(4) (b) g(-2) = 3 = maximum value of g on [-4,4] (c) g(2) = –1 = minimum value of g on [-4, 4]. (d) g'(0) = 0 (so 0 is a critical number of g), but g(0) is not an extreme value of g.
Consider a function z = F(u(s, t), v(s, t)) where F, u, and u are differentiable and u(8, 6) = -1, v(8, 6) = 2, u, (8, 6) = 5, u, (8, 6) = 9, vs(8, 6) = −3, v₁(8, 6) = 4, Fu(-1,2) = -4, and F(-1,2)= -10. Compute and when s = 8 and t = 6 dt
30-

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5. Assume f: [0, 1] → [0, 1] is continuous with f(0) = f(1). Prove there are x, y = [0, 1] with |x - y = and f(x) = f(y)