Question

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

पर्याय

• e

• `1/e`

• `-1/e`

• `2/e`

• `-e`

Answer 1

\[\frac{- 1}{e}\]

 

\[\text { Here }, \]

\[f\left( x \right) = x \log_e x\]

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

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

\[f'\left( x \right) = 0\]

\[ \Rightarrow \log_e x + 1 = 0\]

\[ \Rightarrow \log_e x = - 1\]

\[ \Rightarrow x = e^{- 1} \]

\[\text { Now,} \]

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

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

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

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

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