(a) Use the Mean-Value Theorem to show that if f isdifferentiable on an interval, and if

Chapter 4, Problem 27

(choose chapter or problem)

(a) Use the Mean-Value Theorem to show that if f is differentiable on an interval, and if \(\left|f^{\prime}(x)\right| \leq M\) for all values of x in the interval, then

\(|f(x)-f(y)| \leq M|x-y|\)

for all values of x and y in the interval.

(b) Use the result in part (a) to show that

\(|\sin x-\sin y| \leq|x-y|\)

for all real values of x and y.

Equation Transcription:

Text Transcription:

|f'(x)| less than or equal to M

|f(x)-f(y)| less than or equal to M|x-y|

|sin x-sin y|less than or equal to|x-y|