The minimum value of the function f(x)=x x is - Vidyadip

MCQ

The minimum value of the function $f(x)=x \log x$ is

✓
$-\frac{1}{e}$
B
$-e$
C
$\frac{1}{e}$
D
$e$

✓

Answer

Correct option: A.

(a): We have $f(x)=x \log x \Rightarrow f^{\prime}(x)=x \times \frac{1}{x}+\log x \times 1$
$
\Rightarrow f^{\prime}(x)=1+\log x
$
Now, $f^{\prime}(x)=0 \Rightarrow 1+\log x=0 \Rightarrow \log _e x=-1$
$
\Rightarrow x=e^{-1}=\frac{1}{e}
$
Also $f^{\prime \prime}(x)=\frac{1}{x} \therefore f^{\prime \prime}\left(\frac{1}{e}\right)=\frac{1}{1 / e}=e>0$
$\therefore f(x)$ is minimum at $x=\frac{1}{e}$
and Minimum value $=f\left(\frac{1}{e}\right)=\frac{1}{e} \log \frac{1}{e}=\frac{-1}{e}$

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