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Let f: RR be a differentiable function then for any a, b E R, ab, the Lagrange's theorem says that f(b) f(a) = f'(c)(b-a), where is some number between a and b. Using this fact, prove that if f(x) is bounded, namely f'(c) < M, for some M> 0 and for all then f is a Lipschitz function. 2. Using the above result, prove that sin r. cos r. arctan z. ln(12²) is Lipschitz function.

Let f: RR be a differentiable function then for any a, b E R, ab, the Lagrange's theorem says that f(b) f(a) = f'(c)(b-a), where is some number between a and b. Using this fact, prove that if f(x) is bounded, namely f'(c) < M, for some M> 0 and for all then f is a Lipschitz function. 2. Using the above result, prove that sin r. cos r. arctan z. ln(12²) is Lipschitz function.

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter8: Further Techniques And Applications Of Integration
Section8.2: Integration By Parts
Problem 44E
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Let f: RR be a differentiable function then for any a, b E R, ab, the Lagrange's theorem says that f(b) f(a) = f'(c)(b − a), where is some number between a and b. Using this fact, prove that if f(x) is bounded, namely f'(c) < M, for some M> 0 and for all then f is a Lipschitz function. 2. Using the above result, prove that sin r. cos r. arctan z. ln(1 +2²) is Lipschitz function.
Transcribed Image Text:Let f: RR be a differentiable function then for any a, b E R, ab, the Lagrange's theorem says that f(b) f(a) = f'(c)(b − a), where is some number between a and b. Using this fact, prove that if f(x) is bounded, namely f'(c) < M, for some M> 0 and for all then f is a Lipschitz function. 2. Using the above result, prove that sin r. cos r. arctan z. ln(1 +2²) is Lipschitz function.
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