### Home > APCALC > Chapter 6 > Lesson 6.1.2 > Problem6-29

6-29.

Let $f(x)=\left\{ \begin{array} { c c } { 3 x ^ { 2 / 3 } } & { \text { for } x \leq 1 } \\ { a + b x ^ { 2 } } & { \text { for } x > 1 } \end{array} \right.$ where $a$ and $b$ are constants. .

1. For what value(s) of $x$ is the graph non-differentiable, regardless of the values of $a$ and $b$? Explain what happens to the graph at these points.

$f(x)$ has a cusp within the domain of one of the pieces. Where is it?

2. Determine the values of $a$ and $b$ such that the graph is both continuous and differentiable at $x = 1$.

Write a system of equations: $[\text{piece 1 of } f\left(x\right)] = [\text{piece 2 of } f\left(x\right)],$ evaluated at $x = 1$. $[\text{piece 1 of }f '\left(x\right)] = [\text{piece 2 of }f '\left(x\right)]$, evaluated at $x = 1$.
Use algebra to solve for $a$ and $b$.

Use the eTool below to visualize the problem.
Click the link at right for the full version of the eTool: Calc 6-29 HW eTool