Question

Find the values of m and b that make the following function differentiable.

the piecewise function f of x equals x squared when x is less than or equal to three or mx plus b when x is greater than three

Answer 1

For $f$ to be differentiable, it must be continuous, so we need to have

$\displaystyle\lim_{x\to3^-}f(x)=\lim_{x\to3^+}f(x)=f(3)$

By its definition, $f(3)=3^2=9$. The one-sided limits are

$\displaystyle\lim_{x\to3^-}f(x)=\lim_{x\to3}x^2=9$

$\displaystyle\lim_{x\to3^+}f(x)=\lim_{x\to3}mx+b=3m+b$

so we require $3m+b=9$.

In order for $f$ to be differentiable at $x=3$, we also need to have $f'(3)$ exist, which requires that $f'$ also be continuous at $x=3$. First, compute the derivatives of all pieces of $f$:

$f'(x)=\begin{cases}2x&\text{for }x<3\\?&\text{for }x=3\\m&\text{for }x>3\end{cases}$

$f'$ is continuous at $x=3$ if

$\displaystyle\lim_{x\to3^-}f'(x)=\lim_{x\to3^+}f'(x)=f'(3)$

The one-side limits are

$\displaystyle\lim_{x\to3^-}f'(x)=\lim_{x\to3}2x=6$

$\displaystyle\lim_{x\to3^+}f'(x)=\lim_{x\to3}m=m$

so we need to have $m=6$, and moreover $f$ will be differentiable if we set $f'(3)=6$.

So with $m=6$, we must have $3m+b=9\implies b=-9$.