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Question
If \[f\left( x \right) = \sqrt{x^2 - 10x + 25}\] then the derivative of f (x) in the interval [0, 7] is ____________ .
Options
1
-1
0
none of these
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Solution
none of these
\[\text { We have, } f\left( x \right) = \sqrt{x^2 - 10x + 25}\]
\[ = \sqrt{\left( x - 5 \right)^2}\]
\[ = \left| x - 5 \right| \]
`={[x-5 " for " x>5],[-(x-5) " for " x<5]:}`
\[\text { LHD }= \lim_{x \to 5^-} \frac{f\left( x \right) - f\left( a \right)}{x - a}\]
\[ = \lim_{x \to 5^-} \frac{\sqrt{x^2 - 10x + 25} - \sqrt{5^2 - 10\left( 5 \right) + 25}}{x - 5}\]
\[ = \lim_{x \to 5^-} \frac{\left| x - 5 \right|}{x - 5}\]
\[ = \lim_{x \to 5^-} \frac{- \left( x - 5 \right)}{x - 5}\]
\[ = - 1\]
\[RHD = \lim_{x \to 5^+} \frac{f\left( x \right) - f\left( a \right)}{x - a}\]
\[ = \lim_{x \to 5^+} \frac{\sqrt{x^2 - 10x + 25} - \sqrt{5^2 - 10\left( 5 \right) + 25}}{x - 5}\]
\[ = \lim_{x \to 5^+} \frac{\left| x - 5 \right|}{x - 5}\]
\[ = \lim_{x \to 5^+} \frac{x - 5}{x - 5}\]
\[ = 1\]
\[\text { Here, LHD } \neq RHD\]
\[\text { Thus, the function is not differentiable at }x = 5\]
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