4. Let f be a function continuous on [0, 1] and twice differentiable on (0,1). (a) Suppose that = 0 and f(c) > 0 for some c € (0, 1). f(0) = f(1) = 0 and Prove that there exists xo € (0, 1) such that f"(ro) < 0. (b) Suppose that f* f(x) dx = f(0) = f(1) = 0. Prove that there exists a number xo € (0, 1) such that f"(xo) = 0.
4. Let f be a function continuous on [0, 1] and twice differentiable on (0,1). (a) Suppose that = 0 and f(c) > 0 for some c € (0, 1). f(0) = f(1) = 0 and Prove that there exists xo € (0, 1) such that f"(ro) < 0. (b) Suppose that f* f(x) dx = f(0) = f(1) = 0. Prove that there exists a number xo € (0, 1) such that f"(xo) = 0.
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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![4. Let f be a function continuous on [0, 1] and twice differentiable on (0, 1).
(a) Suppose that
= 0 and f(c) > 0 for some c € (0, 1).
f(0) = f(1) = 0 and
Prove that there exists xo € (0, 1) such that f"(xo) < 0.
(b) Suppose that
f* f(x) dx = f(0) = f(1) = 0.
Prove that there exists a number xo € (0, 1) such that f"(xo) = 0.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6fd6ec40-fd15-430c-9f06-bb8ccef3f331%2F52008607-71ab-4398-a3e7-b5a231aca73b%2F12rk33_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4. Let f be a function continuous on [0, 1] and twice differentiable on (0, 1).
(a) Suppose that
= 0 and f(c) > 0 for some c € (0, 1).
f(0) = f(1) = 0 and
Prove that there exists xo € (0, 1) such that f"(xo) < 0.
(b) Suppose that
f* f(x) dx = f(0) = f(1) = 0.
Prove that there exists a number xo € (0, 1) such that f"(xo) = 0.
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