Let's find values $of$ a and $b$ such that the piecewise defined function

$f(x)=\{\begin{array}{cc}{x}^3+4,& x1\\ ax+b,& x>1\end{array}$

$is$ differentiable. First $we$ make the function continuous:

$\underset{x{1}^{-}}{lim}f(x)=$

$\underset{x{1}^{+}}{lim}f(x)=$

$*f(1)=$

$.$

These three quantities are equal $if$ a and $b$ satisfy the condition:

$$

Now $we$ make the slopes match:

$\underset{x{1}^{-}}{lim}{f}^{\text{'}}(x)=$

$\underset{x{1}^{+}}{lim}{f}^{\text{'}}(x)=$

$=$

$So$ the two slopes are equal when

$a=$

Substitute this into the previous condition $to$ find that

$b=$