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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
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.]
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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