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

6-29.

Given: $f ( x ) = \left\{ \begin{array} { c l } { 3 x ^ { 2 / 3 } } & { \text { for } x \leq 1 } \\ { a + b x ^ { 2 } } & { \text { for } x > 1 } \end{array} \right.$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 $f(x)$ at these points.

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

2. Find values of $a$ and $b$ so the graph is both continuous and differentiable at $x = 1$.

Write a system of equations: [piece $1$ of $f(x)$] $=$ [piece $2$ of $f(x)$], evaluated at $x = 1$. [piece $1$ of $f^\prime(x)$] $=$ [piece $2$ of $f^\prime(x)$], 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