Q. 13 - Calculus

Question

Sketch secant lines on a graph of $f\left(x\right)=\left|x\right|$, and use them to argue that the absolute value function is not differentiable at $x=0$.

Step-by-Step Solution

Answer

The secant lines on a graph are as follows

Step 1. Given information

We have to sketch the secant on a graph of $f\left(x\right)=\left|x\right|$ and by using them, tell that function is not differentiable at $x=0$.

Step 2. Draw the graph for the function

$f\left(x\right)=\left|x\right|$ can be written as

$f\left(x\right)=\left\{\begin{array}{l}x, if x\ge 0\\ -x, if x<0\end{array}\right.$

So, the graph of the function is

Step 3. Draw secant lines and prove graph is not differentiable at x = 0

The secant lines are

From secant lines it is observed that at $x=0$, the slope of the tangent lines approaches to $-1$ from left and $1$ from right. Therefore, the function is not differentiable at $x=0$.