Monotonicity

The function $f'(x)$ is

Match the statements/expressions in Column I with the open intervals in Column II.

For the function
$$f(x)=x\cos\frac1x,\qquad x\ge1,$$

Match the statements/expressions given in Column I with the values given in Column II.

Let the function $g:(-\infty,\infty)\to\left(-\frac\pi2,\frac\pi2\right)$ be given by $g(u)=2\tan^{-1}(e^u)-\frac\pi2$. Then, $g$ is

Which of the following is true?

Let
$$g(x)=\int_0^{e^x}\frac{f'(t)}{1+t^2}dt.$$
Which of the following is true?

In the following $[x]$ denotes the greatest integer less than or equal to $x$.

Match the functions in Column I with the properties in Column II and indicate your answer by darkening the appropriate bubbles in the 4 × 4 matrix given in the ORS.

The line $y=x$ meets $y=ke^x$ for $k\le0$ at