Question

The function $f(x)=x^2\log x$ in the interval $[1,e]$ has

• A. a point of maximum and minimum
• B. a point of maximum only
• C. no point of maximum and minimum in $[1,e]$
• D. no point of maximum and minimum

Answer 1

Answer: (C)

Given $f(x)=x^2\log x$
Now for maximum and minimum $f^{\prime }(x)=0$
$\Rightarrow \frac{x^2}{x}+2x\log x=0$
$\Rightarrow x(1+2\log x)=0$
$\Rightarrow$ either $x=0$, or
$\log x=\frac{-1}{2}$ or $x=e^{-1/2}$
But we have to search the minima and maxima in the interval $[1,e]$
$\therefore f^{\prime \prime }(x)=x\cdot \frac{2}{x}+1+2\log x=3+2\log x$
Now at $x=1f^{\prime \prime }(1)=3=+ve$
$\Rightarrow f(x)$ is $min$. when $x=1$
Therefore min. value $=1^2\log 1=0$
Now at $x=$ e $f^{\prime \prime }(e)=3+2=5(\because$ lne $=1)$
this is also $+ve$
But this is not possible that function has two min. values for two diff. values of $x$. Thus, the function $f(x)$
has no point of maximum and minimum in the interval $[1,e]$