Question

Write the number of points where f(x) = |x+2|+|x+3|
is not differentiable.

Answer 1

$\large\underline{\sf{Solution-}}$

Given function is

$\rm :\longmapsto\:f(x) = |x + 2| + |x + 3|$

Let first define the function f(x).

Here, we have two critical points, -2 and -3.

So,

$\begin{gathered}\begin{gathered}\bf\: f(x) = \begin{cases} &\sf{ - (x + 2) - (x + 3) \: \: \: when \: x < - 3} \\ &\sf{(x + 3) - (x + 2) \: \: \: when \: - 3 \leqslant x \leqslant - 2}\\ &\sf{x + 2 + (x + 3) \: \: \: when \: x > - 2 } \end{cases}\end{gathered}\end{gathered}$

can be further simplified to

$\begin{gathered}\begin{gathered}\bf\: f(x) = \begin{cases} &\sf{ - 2x - 5 \: \: \: when \: x < - 3} \\ &\sf{ \: \: \: \: \: \: 1 \: \: \: when \: - 3 \leqslant x \leqslant - 2}\\ &\sf{2x + 5 \: \: \: when \: x > - 2 } \end{cases}\end{gathered}\end{gathered}$

Let we check differentiability at breaking points

Differentiability at x = - 2

Left Hand Derivative

$\rm :\longmapsto\:\displaystyle\lim_{x \to - 2^{ - } } \frac{f(x) - f( - 2)}{x - ( - 2)}$

$\rm \:  =  \: \displaystyle\lim_{x \to - 2^{ - } } \frac{1 - 1}{x + 2}$

$\rm \:  =  \: 0$

Right Hand Derivative

$\rm :\longmapsto\:\displaystyle\lim_{x \to - 2^{ + } } \frac{f(x) - f( - 2)}{x - ( - 2)}$

$\rm \:  =  \: \displaystyle\lim_{x \to - 2^{ + } } \frac{2x + 5 - 1}{x + 2}$

$\rm \:  =  \: \displaystyle\lim_{x \to - 2^{ + } } \frac{2x + 4}{x + 2}$

$\rm \:  =  \: \displaystyle\lim_{x \to - 2^{ + } } \frac{2(x + 2)}{x + 2}$

$\rm \:  =  \: \displaystyle\lim_{x \to - 2^{ + } } 2$

$\rm \:  =  \: 2$

$\bf\implies \:LHD \: \ne \: RHD$

$\rm \implies\:f(x) \: is \: not \: differentiable \: at \: x = - 2$

Differentiability at x = - 3

Left Hand Derivative

$\rm :\longmapsto\:\displaystyle\lim_{x \to - 3^{ - } } \frac{f(x) - f( - 3)}{x - ( - 3)}$

$\rm \:  =  \: \displaystyle\lim_{x \to - 3^{ - } } \frac{ - 2x - 5 - 1}{x + 3}$

$\rm \:  =  \: \displaystyle\lim_{x \to - 3^{ - } } \frac{ - 2x - 6}{x + 3}$

$\rm \:  =  \: \displaystyle\lim_{x \to - 3^{ - } } \frac{ - 2(x + 3)}{x + 3}$

$\rm \:  =  \: \displaystyle\lim_{x \to - 3^{ - } } - 2$

$\rm \:  =  \: - 2$

Right Hand Derivative

$\rm :\longmapsto\:\displaystyle\lim_{x \to - 3^{ + } } \frac{f(x) - f( - 3)}{x - ( - 3)}$

$\rm \:  =  \: \displaystyle\lim_{x \to - 3^{ - } } \frac{1 - 1}{x + 3}$

$\rm \:  =  \: 0$

$\bf\implies \:LHD \: \ne \: RHD$

$\rm \implies\:f(x) \: is \: not \: differentiable \: at \: x = - 3$

Hence,

$\rm \implies\:f(x) \: is \: not \: differentiable \: at \: x = - 3 \: and \: x = - 2$

$\rm \implies\:f(x) \: is \: not \: differentiable \: exactly \: at \: 2 \: points$