Question

the set of points Where the function f geven by f(x) =|x-3| cosn is different able is​

Answer 1

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

Given function is

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

Let us first define the function f(x)

We know

$\begin{gathered}\begin{gathered}\bf\: \rm :\longmapsto\: |x| = \begin{cases} &\sf{ - x, \: when \: x < 0} \\ &\sf{ \: \: \: x, \: when \: x \geqslant 0} \end{cases}\end{gathered}\end{gathered}$

So,

$\begin{gathered}\begin{gathered}\bf\: \rm :\longmapsto\:f(x) = |x - 3| = \begin{cases} &\sf{ - (x - 3), \: when \: x < 3} \\ &\sf{ \: \: \: x - 3, \: when \: x \geqslant 3} \end{cases}\end{gathered}\end{gathered}$

So, we have to check differentiability of f(x) at x = 3.

We know,

A function f(x) is said to be differentiable at x = a iff

$\boxed{ \rm{ \displaystyle\lim_{x \to a^-} \dfrac{f(x) - f(a)}{x - a} = \displaystyle\lim_{x \to a^ + } \dfrac{f(x) - f(a)}{x - a}}}$

Consider, Left Hand Derivative at x = 3

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

$\rm \:  =  \:  \: \displaystyle\lim_{x \to 3^-} \dfrac{ - (x - 3) -0}{x - 3}$

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

$\rm \:  =  \:  \: \displaystyle\lim_{x \to 3^-} ( - 1)$

$\rm \:  =  \:  \: - 1$

$\rm :\longmapsto\:\displaystyle\lim_{x \to 3^-} \dfrac{f(x) - f(3)}{x - 3} = - 1$

Now, Consider Right Hand Derivative at x = 3

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

$\rm \:  =  \:  \: \displaystyle\lim_{x \to 3^ + } \dfrac{ (x - 3) -0}{x - 3}$

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

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

$\rm \:  =  \:  \: 1$

$\rm :\longmapsto\:\displaystyle\lim_{x \to 3^ + } \dfrac{f(x) - f(3)}{x - 3} = 1$

Thus, we concluded that

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

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

$\bf\implies \:f(x) \: is \: differentiable \: \forall \: x \in \: R - \{3 \}$