Question

$f\left(x\right)=x^x$ is decreasing in ________ interval, $x\in R^{+}$.

• A. $(0,\frac{1}{e})$
• B. $(0,e)$
• C. $(0,1)$
• D. $(0,\infty )$

Answer 1

Answer: (A)

$(0,\frac{1}{e})$

$f\left(x\right)=x^x$
Now taking $y=x^x$ $(\because y=f(x\left)\right)$
$\therefore logy=xlogx$
Now differentiate w.r. to x
$\therefore \frac{1}{y}\frac{dy}{dx}=x\left(\frac{1}{x}\right)+1\cdot logx$
$\therefore \frac{dy}{dx}=y(1+logx)$
$=x^x(1+logx)$
Hence $f\left(x\right)=x^x(1+logx)$
Now, $f\left(x\right)$ is decreasing function.
$\therefore f^{\prime }\left(x\right)<0$
$\therefore x^x(+logx)<0$
$\therefore 1+logx<0$
$\therefore logx<-1$
$\therefore x<e^{-1}$
$\therefore x<\frac{1}{e}$
$\therefore x\in (0,\frac{1}{e})$ function $f\left(x\right)=x^x$ is decreasing function.