Chapter 2.7, Problem 51E

To determine

To show: The function $f\left(x\right)=\left|x-6\right|$ is not differentiable at $x=6$ and the formula of derivative of $f\left(x\right)$.

Explanation of Solution

Definition used:

An absolute function $\left|x\right|$ is defined as follows.

$\left|x\right|=\{\begin{array}{cc}-x& \text{ if }x<0\\ x& \text{ if }x\ge 0\end{array}$

Proof:

Rewrite the function $f\left(x\right)$ in terms of a piecewise function.

Use the definition of absolute value function to define the function $\left|x-6\right|$,

$\begin{array}{c}f\left(x\right)=\{\begin{array}{cc}-\left(x-6\right)& \text{if }x-6<0\\ x-6& \text{if }x-6\ge 0\end{array}\\ =\{\begin{array}{cc}-x+6& \text{if }x<6\\ x-6& \text{if }x\ge 6\end{array}\end{array}$

The left hand derivative at $x=6$ is calculated as follows,

${f^{\prime }}_{-}\left(6\right)=\underset{x\to 6^{-}}{\lim }\frac{f\left(x\right)-f\left(6\right)}{x-6}$

Since $x$ tends to $6^{-}$,$x<6$

$\begin{array}{c}{f^{\prime }}_{-}\left(6\right)=\underset{x\to 6^{-}}{\lim }\frac{-x+6-\left(-6+6\right)}{x-6}\\ =\underset{x\to 6^{-}}{\lim }\frac{-x+6}{x-6}\\ =\underset{x\to 6^{-}}{\lim }\frac{-\left(x-6\right)}{\left(x-6\right)}\\ =-1\end{array}$

Thus, ${f^{\prime }}_{-}\left(6\right)=-1$. (1)

The right hand derivative at $x=6$ is calculated as follows,

${f^{\prime }}_{+}\left(6\right)=\underset{x\to 6^{+}}{\lim }\frac{f\left(x\right)-f\left(6\right)}{x-6}$

Since $x$ tends to $6^{+}$,$x>6$

$\begin{array}{c}{f^{\prime }}_{+}\left(6\right)=\underset{x\to 6^{+}}{\lim }\frac{x-6-\left(6-6\right)}{x-6}\\ =\underset{x\to 6^{+}}{\lim }\frac{x-6}{x-6}\\ =1\end{array}$

Thus, ${f^{\prime }}_{+}\left(6\right)=1$. (2)

From the equation (1) and (2), ${f^{\prime }}_{-}\left(6\right)\ne {f^{\prime }}_{+}\left(6\right)$.

This implies that, the value of $f^{\prime }\left(6\right)$ does not exist.

Therefore, the function $f\left(x\right)=\{\begin{array}{cc}-x+6& \text{if }x<6\\ x-6& \text{if }x\ge 6\end{array}$ is not differentiable at $x=6$

Hence, the required proof is obtained.

Now find the formula of derivative of $f\left(x\right)$.

From equation (1) and (2), ${f^{\prime }}_{-}\left(6\right)=-1$ and ${f^{\prime }}_{+}\left(6\right)=1$.

This implies that, the function $f^{\prime }\left(x\right)=-1$ at $x<6$ and $f^{\prime }\left(x\right)=1$ at $x>6$.

Thus the required derivative function is, $f^{\prime }\left(x\right)=\{\begin{array}{cc}-1& \text{if }x<6\\ 1& \text{if }x>6\end{array}$.

Graph:

Use the above information to draw the graph of $f^{\prime }$ as shown below in Figure 1.

Thus,$f^{\prime }$ is the required graph.