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# The Minimum Value of X Log E X is (A) E (B) 1/E (C) 1 (D) None of These - Mathematics

#### Question

The minimum value of $\frac{x}{\log_e x}$ is _____________ .

##### Options
• e

• 1/e

• 1

• none of these

#### Solution

e

$\text { Given }: f\left( x \right) = \frac{x}{\log_e x}$

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

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

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

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

$\Rightarrow \log_e x - 1 = 0$

$\Rightarrow \log_e x = 1$

$\Rightarrow x = e$

$\text { Now,}$

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

$\Rightarrow f''\left( e \right) = \frac{- 1}{e} + \frac{2}{e} = \frac{1}{e} > 0$

$\text { So, x = e is a local minima } .$

$\therefore \text { Minimum value of } f\left( x \right) = \frac{e}{\log_e e} = e$

Is there an error in this question or solution?