5. Prove that there is no value of k such that the equation æ³ – 3x + k = 0 has two distinct roots in [0, 1]. Hint: Let f(x) = x³ – 3+ k. Assume there are two distinct a, b e (0, 1] such that f(a) = f(b) = 0. Apply Rolle's theorem to f on the interval (a, b] to conclude that there exists a c such that f'(c) = 0. This should lead to a contradiction.
5. Prove that there is no value of k such that the equation æ³ – 3x + k = 0 has two distinct roots in [0, 1]. Hint: Let f(x) = x³ – 3+ k. Assume there are two distinct a, b e (0, 1] such that f(a) = f(b) = 0. Apply Rolle's theorem to f on the interval (a, b] to conclude that there exists a c such that f'(c) = 0. This should lead to a contradiction.
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. Prove that there is no value of k such that the equation x³ – 3x + k = 0 has two distinct
roots in [0, 1]. Hint: Let f(x) = x³ – 3 + k. Assume there are two distinct a, b e [0, 1] such
that f(a) = f(b) = 0. Apply Rolle's theorem to f on the interval [a, 6] to conclude that there
exists a c such that f'(c) = 0. This should lead to a contradiction.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffbb66fa7-7c22-4982-a22f-aaed542f65b3%2F8aa78b70-0c1c-46f2-9bbe-d139d0a56c99%2Fqkanqii_processed.png&w=3840&q=75)
Transcribed Image Text:5. Prove that there is no value of k such that the equation x³ – 3x + k = 0 has two distinct
roots in [0, 1]. Hint: Let f(x) = x³ – 3 + k. Assume there are two distinct a, b e [0, 1] such
that f(a) = f(b) = 0. Apply Rolle's theorem to f on the interval [a, 6] to conclude that there
exists a c such that f'(c) = 0. This should lead to a contradiction.
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