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# The Minimum Value of X Loge X is Equal to (A) E (B) 1/E (C) − 1/E (D) 2/E (E) − E - CBSE (Science) Class 12 - Mathematics

#### Question

The minimum value of x loge x is equal to ____________ .

• e

• 1/e

• -1/e

• 2/e

• -e

#### Solution

$\frac{- 1}{e}$

$\text { Here },$

$f\left( x \right) = x \log_e x$

$\Rightarrow f'\left( x \right) = \log_e x + 1$

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

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

$\Rightarrow \log_e x + 1 = 0$

$\Rightarrow \log_e x = - 1$

$\Rightarrow x = e^{- 1}$

$\text { Now,}$

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

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

$\text { So,} x = e^{- 1}\text { is a local minima }.$

$\text { Hence, the minimum value of } f\left( x \right) = f\left( e^{- 1} \right) .$

$\Rightarrow e^{- 1} \log_e \left( e^{- 1} \right) = - e^{- 1} = \frac{- 1}{e}$

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Solution The Minimum Value of X Loge X is Equal to (A) E (B) 1/E (C) − 1/E (D) 2/E (E) − E Concept: Graph of Maxima and Minima.
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