Question

Find the point of maxima and minima of $f\left(x\right)=x\ln x$

• A. local min. at $x=\left(\frac{1}{e}\right),$ No local max
• B. local max. at $x=\left(\frac{1}{e}\right),$ No local min
• C. No local max and min
• D. none of these

Answer 1

Answer: (A)

local min. at $x=\left(\frac{1}{e}\right),$ No local max

$y=xln\left(x\right)$
$y^{\prime }=ln\left(x\right)+1$
Now
$y^{\prime }=0$ implies
$ln\left(x\right)=-1$
Or
$x=e^{-1}$.
$y"\left(x\right)=\frac{1}{x}$
Hence
$y"\left(e^{-1}\right)>0$.. Hence minima.
Thus
$y\left(e^{-1}\right)=-\left(e^{-1}\right)$.
Hence the minimum value of $y=xln\left(x\right)$ is $\frac{-1}{e}$ and $x=\frac{1}{e}$.