Use the Mean-Value Theorem to prove the following result:Let f be continuous at x0 and

Chapter 4, Problem 40

(choose chapter or problem)

Use the Mean-Value Theorem to prove the following result: Let \(f\) be continuous at \(x_{0}\) and suppose that \(\lim _{x \rightarrow x_{0}} f^{\prime}(x)\) exists. Then \(f\) is differentiable at \(x_{0}\), and

\(f^{\prime}\left(x_0\right)=\lim_{x\rightarrow x_0}f^{\prime}(x)\)

[Hint: The derivative\(f^{\prime}\left(x_{0}\right)\) is given by

\(f^{\prime}\left(x_{0}\right)=\lim _{x \rightarrow x_{0}} \frac{f(x)-f\left(x_{0}\right)}{x-x_{0}}\)

provided this limit exists.]

Equation Transcription:

Text Transcription:

f

x_0

lim_x right arrow x_0 f prime (x)

f

x_0

f prime (x_0)=lim_x right arrow x_0 f prime (x)

f prime (x_0)

f prime (x_0)=lim_x right arrow x_0 frac f(x)-f(x_0)/x-x_0