Answer by appropriately matching the information given in the three columns of the following table.

Question

Answer  by appropriately matching the information given in the three columns of the following table. \Let $f(x)=x+\log _{e} x-x \log _{e} x, x \in(0, \infty)$ \Column 1 contains information about zeros of $f(x), f^{\prime}(x)$ and $f^{\prime \prime}(x)$. \Column 2 contains information about the limiting behaviour of $f(x), f^{\prime}(x)$ and $f^{\prime \prime}(x)$ at infinity. \Column 3 contains information about increasing / decreasing nature of $f(x)$ and $f^{\prime}(x)$. \Which of the following options is the only correct combination?\

Professor PC Thomas Classes · Answer

(a) : Note that $f(x)=x+\ln x-x \ln x$\ $\lim _{x ightarrow \infty} f(x)=\lim _{x ightarrow \infty} 1-[(x-1)(\ln x-1)]=-\infty$\ $f^{\prime}(x)=1+\frac{1}{x}-\ln x-1=\frac{1}{x}-\ln x$\ $f^{\prime \prime}(x)=-\frac{1}{x^{2}}-\frac{1}{x}<0$\ (II) is true as $f^{\prime}(1)=1-0=1$\ $f^{\prime}(e)=\frac{1}{e}-1<0$\ (ii) is true.\ (Q) is true as $f^{\prime}(x)<0$ in $\left(e, e^{2}ight)$.\