MCQ
Local maximum value of the function ${{\log x} \over x}$ is
- A$e$
- B$1$
- ✓${1 \over e}$
- D$2e$
For maximum or minimum value of $f(x),\,\,f'(x) = 0$
==> $f'(x) = \frac{{1 - {{\log }_e}x}}{{{x^2}}} = 0$ or $\frac{{1 - {{\log }_e}x}}{{{x^2}}} = 0$
$\therefore {\log _e}x = 1$ or $x = e$, which lie in $(0,\infty )$.
For $x = e,\,\,\frac{{{d^2}y}}{{d{x^2}}} = - \frac{1}{{{e^3}}}$, which is $ - ve$.
Hence $ y $ is maximum at $x = e$ and its maximum value $ = \frac{{\log e}}{e} = \frac{1}{e}$.
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| Column $I$ | Column $II$ |
| $(A)$ The minimum value of $\frac{x^2+2 x+4}{x+2}$ is | $(p)$ $0$ |
| $(B)$ Let $A$ and $B$ be $3 \times 3$ matrices of real numbers, where $A$ is symmetric, $B$ is skewsymmetric, and $(A+B)(A-B)=(A-B)(A+B)$. If $(A B)^t=(-1)^k A B$, where $(A B)^t$ is the transpose of the matrix $A B$, then the possible values of $k$ are | $(q)$ $1$ |
| $(C)$ Let $\mathrm{a}=\log _3 \log _3 2$. An integer $\mathrm{k}$ satisfying $1<2^{\left(-k+3^{-2}\right)}<2$, must be less than | $(r)$ $2$ |
| $(D)$ If $\sin \theta=\cos \phi$, then the possible values of $\frac{1}{\pi}\left(\theta \pm \phi-\frac{\pi}{2}\right)$ are | $(s)$ $3$ |