Let f: R → R be a differentiable function then for any a, b = R, ab, the Lagrange's theorem says that f(b) f(a) = f'(c)(b-a), where c is some number between a and b. Using this fact, prove that if f'(r) is bounded, namely |f'(c)| ≤ M, for some M > 0 and for all c ER then f is a Lipschitz function. Using the above result, prove that sin r, cos x, arctan r, ln(1 + r²) is Lipschitz function.

Let f: R → R be a differentiable function then for any a, b = R, ab, the Lagrange's theorem says that f(b) f(a) = f'(c)(b-a), where c is some number between a and b. Using this fact, prove that if f'(r) is bounded, namely |f'(c)| ≤ M, for some M > 0 and for all c ER then f is a Lipschitz function. Using the above result, prove that sin r, cos x, arctan r, ln(1 + r²) is Lipschitz function.

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

See similar textbooks

Similar questions

Determine if the statemment is true or false. If the statement is false, then correct it and make it true. If the function f increases on the interval -,x1 and decreases on the interval x1,, then fx1 is a local minimum value.
Find two nonnegative number x and y for which x+3y=30, such that x2y is maximized.
What does it mean to say that f(4) is a local maximum valueof f ?
What is the purpose of the Intermediate Value Theorem?