Question

Column 1 contains the equation for 4 different functions
Column 2 contains information about increasing/decreasing nature of f(x)
Column 3 contains information about the limiting behaviour of f(x) near 0 and at infinity
$\begin{array}{cccccc}& ColumnI& & Column2& & Column3\\ \left(I\right)& f\left(x\right)=x^{2}logx& \left(i\right)& f\left(x\right)hasonepointofminima& \left(P\right)& li{m}_{x\to O^{+}}f\left(x\right)=0\\ \left(II\right)& f\left(x\right)=xlogx& \left(ii\right)& f\left(x\right)hasonepointofmaxima& \left(Q\right)& li{m}_{x\to \infty }f\left(x\right)=0\\ \left(III\right)& f\left(x\right)=\frac{logx}{x}& \left(iii\right)& f\left(x\right)increasesin(0,e)& \left(R\right)& li{m}_{x\to \infty }f\left(x\right)=\infty \\ \left(IV\right)& f\left(x\right)=x^{-x}& \left(iv\right)& f\left(x\right)decreasesin(0,\frac{1}{e})& \left(S\right)& li{m}_{x\to O^{+}}f\left(x\right)=1\end{array}$
Which of the following options is the only INCORRECT combination?

• A. (IV)(iv)(S)
• B. (III)(iii)(Q)
• C. (II)(iv)(R)
• D. (I)(i)(P)

Answer 1

The correct option is A (IV)(iv)(S)


$f\left(x\right)=x^{2}logx$
$f^{\prime }\left(x\right)=x(2logx+1)$
$\Rightarrow x=\frac{1}{\sqrt{e}}$

$f\left(x\right)=xlogx$
$f^{\prime }\left(x\right)=1+logx$
$\Rightarrow x=\frac{1}{e}$

$f\left(x\right)=\frac{logx}{x}$
$f^{\prime }\left(x\right)=\frac{1-logx}{x^2}$
$\Rightarrow x=e$

$f\left(x\right)=x^{-x}$
$f^{\prime }\left(x\right)=-x^{-x}(1+logx)$
$\Rightarrow x=\frac{1}{e}$