6568 You are asked in these exercises to determine whether a piecewise-defined function

Chapter 2, Problem 65

(choose chapter or problem)

You are asked in these exercises to determine whether a piecewise-defined function f is differentiable at a value \(x=x_{0}\), where f is defined by different formulas on different sides of \(x_{0}\). You may use without proof the following result, which is a consequence of the Mean-Value Theorem (discussed in Section 4.8). Theorem. 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 \(f^{\prime}\left(x_{0}\right)=\lim _{x \rightarrow x_{0}} f^{\prime}(x)\).

Show that

\(f(x)=\left\{\begin{array}{ll} x^{2}+x+1, & x \leq 1 \\ 3 x, & x>1 \end{array}\right.\)

is continuous at \(x=1\). Determine whether f is differentiable at \(x=1\). If so, find the value of the derivative there. Sketch the graph of f.

Equation Transcription:

{

Text Transcription:

x=x_0

x_0

lim_x rightarrow x_0 f'(x)

f'(x_0)=lim_x rightarrow x_0 f'(x)

f(x)= {3x, x  > 1      x^2+x+1,  x less than or equal to 1

x=1