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APPLICATION OF DERIVATIVES — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsAPPLICATION OF DERIVATIVES5 Marks
Question
$\text{If function f(x) = |x - 3|} + \text{|x - 4|,} $ then show that f (x) is not differentiable at $\text{x = 3 and x = 4}.$
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Answer

$\text{f(x)} = \text{|x - 3}| + |\text{x - 4|}$
$= \begin{cases} \begin{matrix} 7 - 2x, & x<3 \\ 1 ,& \ \ \ \ \ \ \ 3\leq x<4\\ 2x - 7 ,& x \geq 4 \end{matrix} \end{cases}$
$\text{L.H.D at x} = 3 \lim\limits_{ x \rightarrow 3^{-}}\frac{\text{f(x) - f (3)}}{\text{x - 3}}$
$ \lim\limits_{ x \rightarrow 3^{-}}\frac{\text{6 - 2x}}{\text{x - 3}} = -2$
$\text{R.H.D. at x} = \lim\limits_{ x \rightarrow 3^{+}} \frac{\text{f(x) - f (3)}}{\text{x - 3}}$
$= \frac{1 - 1}{\text{x - 3}} = 0$
$\text{L.H.D}\neq\text{R.H.D}\therefore \text{f (x)is not differentiable x = 3}$
$\text{L.H.D at x = 4}\lim\limits_{ x \rightarrow 4^{-}}\frac{\text{f(x) - f(4)}}{\text{x - 4}}$
$= \frac{1 -1}{\text{x - 4}} = 0$
$\text{R.H.D at x} = 4\lim\limits_{ x \rightarrow 4^{+}} \frac{\text{f(x) - f(4)}}{\text{x -4}}$
$\lim\limits_{ x \rightarrow 4^{+}} \frac{\text{2x - 7 - 1}}{\text{x - 4}} = 2$
$\text{L.H.D at x = 4}\neq\text{R.H.D at x = 4}$
$\text{f(x) is not differentiable at x = 4}$

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