Question

Find the intervals of monotonicity of the function

$x^x$ for $x>0$

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

Answer 1

Answer: (C), (D)

The correct options are
C Increasing in $(\frac{1}{e},\infty ).$
D Decreasing in $(0,\frac{1}{e}).$

Let $f\left(x\right)=x^x$ taking log both side $\log f\left(x\right)=x\log x$
$f^{\prime }\left(x\right)=x^x(1+\log x),f^{\prime }\left(x\right)$ +ive when $1+\log x>0$ or $\log x>-1orx>e^{-1}=\frac{1}{e}$

$\therefore$ Increasing in $(\frac{1}{e},\infty ).$
and $f^{\prime }\left(x\right)$ is -ve when $\log x<-1orx<e^{-1}=\frac{1}{e}$ and $x>0$ $\therefore$ Decreasing in $(0,\frac{1}{e}).$