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 $f'( x )=0$
$\Rightarrow \frac{ x ^{2}}{ x }+2 x\, \log\, x =0$
$ \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 $[1, e ]$
$\therefore f ''( x )= x \cdot \frac{2}{ x }+1+2\, \log \,x =3+2 \,\log \,x$
Now at $x=1 f ''(1)=3=+ ve$
$\Rightarrow f ( x )$ is $min$. when $x =1$
Therefore min. value $=1^{2} \log \,1=0$
Now at $x=$ e $f''(e)=3+2=5(\because$ 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 $[1, e ]$