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# Write the Maximum Value of F(X) = Log X X , If It Exists . - Mathematics

#### Question

Write the maximum value of f(x) = $\frac{\log x}{x}$, if it exists .

#### Solution

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

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

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

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

$\Rightarrow \frac{1 - \log x}{x^2} = 0$

$\Rightarrow 1 - \log x = 0$

$\Rightarrow \log x = 1$

$\Rightarrow \log x = \log e$

$\Rightarrow x = e$

$\text { Now,}$

$f''\left( x \right) = \frac{- x - 2x\left( 1 - \log x \right)}{x^4} = \frac{- 3x - 2x \log x}{x^4}$

$\text { At }x = e:$

$f''\left( e \right) = \frac{- 3e - 2e \log e}{e^4} = \frac{- 5}{e^3} < 0$

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

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

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

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