2. Let a > 0 and 6> a. Suppose that f(x) is continuous on [a, b] and differentiable on (a, b). Then, there are three numbers x1, x2, x3 = (a, b) such that f'(x1)=(a+b) f'(x2) 2x2 = (a² + ab + b²) f'(x3) 3x² [Hint: Apply Cauchy Mean Value Theorem three times with g(x) =x and its square and cube respectively.]
2. Let a > 0 and 6> a. Suppose that f(x) is continuous on [a, b] and differentiable on (a, b). Then, there are three numbers x1, x2, x3 = (a, b) such that f'(x1)=(a+b) f'(x2) 2x2 = (a² + ab + b²) f'(x3) 3x² [Hint: Apply Cauchy Mean Value Theorem three times with g(x) =x and its square and cube respectively.]
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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Let a ≥ 0 and b > a. Suppose that f (x) is continuous on [a, b] and differentiable on (a, b). Then, there are
three numbers x1, x2, x3 ∈ (a, b) such tha,
REFER TO PICTURE, PLEASE SOLVE PROOF IN FULL DETAIL
![2. Let a > 0 and 6> a. Suppose that f(x) is continuous on [a, b] and differentiable on (a, b). Then, there are
three numbers x1, x2, x3 = (a, b) such that
f'(x1)=(a+b)
f'(x2)
2x2
=
(a² + ab + b²) f'(x3)
3x²
[Hint: Apply Cauchy Mean Value Theorem three times with g(x) =x and its square and cube respectively.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F643c9073-bfc9-41d3-b50c-3b14187fea75%2F645e6ab8-49af-4b42-b18e-9b4ddf320fc6%2F2enx8b_processed.png&w=3840&q=75)
Transcribed Image Text:2. Let a > 0 and 6> a. Suppose that f(x) is continuous on [a, b] and differentiable on (a, b). Then, there are
three numbers x1, x2, x3 = (a, b) such that
f'(x1)=(a+b)
f'(x2)
2x2
=
(a² + ab + b²) f'(x3)
3x²
[Hint: Apply Cauchy Mean Value Theorem three times with g(x) =x and its square and cube respectively.]
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