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# Write the Maximum Value of F(X) = X1/X. - CBSE (Science) Class 12 - Mathematics

#### Question

Write the maximum value of f(x) = x1/x.

#### Solution

$\text { Given }: \hspace{0.167em} f\left( x \right) = x^\frac{1}{x}$

$\text { Taking log on both sides, we get }$

$\log f\left( x \right) = \frac{1}{x}\log x$

$\text { Differentiating w . r . t . x, we get }$

$\frac{1}{f\left( x \right)}f'\left( x \right) = \frac{- 1}{x^2}\log x + \frac{1}{x^2}$

$\Rightarrow f'\left( x \right) = f\left( x \right)\frac{1}{x^2}\left( 1 - \log x \right)$

$\Rightarrow f'\left( x \right) = x^\frac{1}{x} \left( \frac{1}{x^2} - \frac{1}{x^2}\log x \right) ...........\left( 1 \right)$

$\Rightarrow f'\left( x \right) = x^\frac{1}{x} - 2 \left( 1 - \log x \right)$

$\text { For a local maxima or a local minima, we must have }$

$f'\left( x \right) = 0$

$\Rightarrow x^\frac{1}{x} - 2 \left( 1 - \log x \right) = 0$

$\Rightarrow \log x = 1$

$\Rightarrow x = e$

$\text { Now },$

$f''\left( x \right) = x^\frac{1}{x} \left( \frac{1}{x^2} - \frac{1}{x^2}\log x \right)^2 + x^\frac{1}{x} \left( \frac{- 2}{x^3} + \frac{2}{x^3}\log x - \frac{1}{x^3} \right) = x^\frac{1}{x} \left( \frac{1}{x^2} - \frac{1}{x^2}\log x \right)^2 + x^\frac{1}{x} \left( - \frac{3}{x^3} + \frac{2}{x^3}\log x \right)$

$\text { At }x = e$

$f''\left( e \right) = e^\frac{1}{e} \left( \frac{1}{e^2} - \frac{1}{e^2}\log e \right)^2 + e^\frac{1}{e} \left( - \frac{3}{e^3} + \frac{2}{e^3}\log e \right) = - e^\frac{1}{e} \left( \frac{1}{e^3} \right) < 0$

$\text { So, x = e is a point of local maximum }.$

$\text { Thus, the maximum value is given by }$

$f\left( e \right) = e^\frac{1}{e}$

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Solution Write the Maximum Value of F(X) = X1/X. Concept: Graph of Maxima and Minima.
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