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

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KollegeApply · Accepted Answer

- 1/e 	ext { Here }, f\left( x ight) = x \log_e x \Rightarrow f'\left( x ight) = \log_e x + 1 	ext { For a local maxima or a local minima, we must have } f'\left( x ight) = 0 \Rightarrow \log_e x + 1 = 0 \Rightarrow \log_e x = - 1 \Rightarrow x = e^{- 1} 	ext { Now,} f''\left( x ight) = 1/x \Rightarrow f''\left( e^{- 1} ight) = e > 0 	ext { So,} x = e^{- 1}	ext { is a local minima }. 	ext { Hence, the minimum value of } f\left( x ight) = f\left( e^{- 1} ight) . \Rightarrow e^{- 1} \log_e \left( e^{- 1} ight) = - e^{- 1} = - 1/e

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