Question

The graph of $y=x\ln x$ is

• A.
• B.
• C.
• D.

Answer 1

The correct option is D

$y=x\ln x,$ Domain $:(0,\infty )$

$x=1\Rightarrow y=0$

So, graph intersect, $x$ -axis at $x=1$ $\underset{x\to 0^{+}}{lim}\left(x\ln x\right)=0$ and $\underset{x\to \infty }{lim}\left(x\ln x\right)=\infty$

$y^{\prime }=1+\ln x=0\Rightarrow x=\frac{1}{e}$

$y^{\prime \prime }=\frac{1}{x}$

$y^{\prime \prime }\left(\frac{1}{e}\right)=e>0$
$\Rightarrow x=\frac{1}{e}$ is a point of local minima

Also $y^{\prime \prime }>0,\forall x\in (0,\infty )$

Hence the curve is concave up $\forall x\in (0,\infty )$

So, option $\left(c\right)$ is correct