Question

Let the function $f$ be defined by $f\left(x\right)=x\ln x$, for all $x>0$. Then

• A. $f$ is increasing on $(0,e^{-1})$
• B. $f$ is decreasing on $(0,1)$
• C. The graph of $f$ is concave down for all $x$
• D. The graph of $f$ is concave up for all $x$

Answer 1

The correct option is D The graph of $f$ is concave up for all $x$

$\frac{dy}{dx}=1+\ln x$
$\frac{dy}{dx}=0\Rightarrow x=e^{-1}=\frac{1}{e}$
$\Rightarrow$ It is increasing on $(\frac{1}{e},\infty )$ and decreasing on $(0,\frac{1}{e})$


Clearly, $x=\frac{1}{e}$ gives local minima
and $f\left(\frac{1}{e}\right)=-\frac{1}{e}$
Also, $\frac{d^{2}y}{d{x}^2}=\frac{1}{x}>0(\because x>0)$
$\Rightarrow$ Concave up for all $x>0$