The minimum value of x _ e x is equal to : - Vidyadip

MCQ

The minimum value of $\text{x}\log_{\text{e}}\text{x}$ is equal to :

A
$\text{e}$
B
$\frac{1}{\text{e}}$
✓
$\frac{-1}{\text{e}}$
D
$\text{2}{\text{e}}$

✓

Answer

Correct option: C.

$\frac{-1}{\text{e}}$

Here, $\text{f}(\text{x})=\text{x}\log_{\text{e}}\text{x}$
lmplies that $\text{f}'(\text{x})=\log_{\text{e}}\text{x}+1$
For a local maxima or a local minima, we must have $f\ '(x) = 0$
lmplies that $\log_{\text{e}}\text{x}+1=0$
lmplies that $\log_\text{e}\text{x}=-1$
lmplies that $\text{x}=\text{e}^{-1}$
Now, $\text{f}\ ''(\text{x})=\frac{1}{\text{x}}$
lmplies that $\text{f}''(\text{e}^{-1})=\text{e} > 0$
Threrfore, $\text{f}(\text{e}^{-1})$ is a local minima.
Hence, the minimum value of $\text{f}(\text{x})=\text{f}(\text{e}^{-1})$
lmplies that $\text{e}^{-1}\log_\text{e}(\text{e}^{-1})=-\text{e}^{-1}=\frac{-1}{\text{e}}$

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