#### Need solution for RD Sharma maths class 12 chapter 10 Differentiation exercise Multiple choice question 23

None of these

Hint:

In case of function of one variable it’s a function that doesn’t have finite derivation

Given:

$f(x)=\sqrt{x^{2}-10 x+25}, f^{\prime}(x)=$

Solution:

$f(x)=\sqrt{x^{2}-10 x+25}$

\begin{aligned} &=\sqrt{(x-5)^{2}} \\\\ &=|x-5| \end{aligned}

$f(x)=\left\{\begin{array}{cll} x-5 & \text { for } & x>5 \\\\ -(x-5) & \text { for } & x<5 \end{array}\right.$

$L H D=\lim _{x \rightarrow 5^{-}} \frac{f(x)-f(a)}{x-a}$

$=\lim _{x \rightarrow 5^{-}} \frac{\sqrt{x^{2}-10 x+25}-\sqrt{5^{2}-10(5)+25}}{x-5}$

$=\lim _{x \rightarrow 5^{-}} \frac{|x-5|}{(x-5)}=\lim _{x \rightarrow 5^{-}} \frac{-(x-5)}{x-5}=-1$

$R H D=\lim _{x \rightarrow 5^{+}} \frac{f(x)-f(a)}{x-a}$

$=\lim _{x \rightarrow 5^{+}} \frac{\sqrt{x^{2}-10 x+25}-\sqrt{5^{2}-10(5)+25}}{x-5}$

\begin{aligned} &=\lim _{x \rightarrow 5^{+}} \frac{|x-5|}{x-5} \\\\ &=\lim _{x \rightarrow 5^{+}} \frac{x-5}{x-5} \\\\ &=1 \end{aligned}

$L H D \neq R H D,$ so function is not differentiable .