Chapter 2.3, Problem 70ES

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 $\underset{x\to x_0}{\lim }{f}^{\prime }\left(x\right)$ exists. Then $f$ is differentiable at $x_0$ , and $f^{\prime }\left(x_0\right)={\lim }_x{{}_{\to }}_{x_0}f\text{'}\left(x\right).$

Let

$f\left(x\right)=\left\{\begin{array}{cc}{x}^3+\frac{1}{16},& x<\frac{1}{2}\\ \frac{3}{4}{x}^2,& x\ge \frac{1}{2}\end{array}\right.$

Determine whether $f$ is differentiable at $x=\frac{1}{2}$ . If so, find the value of the derivative there.