(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|
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