The rate of change of $\sqrt{x^2-2}$ with respect to $\dfrac{x}{x+1}$ at $x=2$ is :

The largest open interval, in which the function $f(x) = \frac{x}{x^2 + 1}$ increases, is

For the function $f(x) = x^{1/x}$, $x > 0$, which of the following are correct?

(A) $x = 0$ is the only point where extremum may occur.

(B) The given function is maximum at $x = e$.

(C) The function has no extreme value for $x > 0$.

(D) The maximum value of the function $f(x)$ is $e^{1/e}$.

Choose the correct answer from the options given below:

If $f(x) = x^3 + px^2 + qx - 3$ has a local maxima at $x = 0$ and local minima at $x = 1$, then the values of $p$ and $q$ respectively, are:

The point on the curve y = (x - 2)² at which the tangent is parallel to the chord joining the points (2, 0) and (4, 4) is:

Match List-I with List-II

Choose the correct answer from the options given below:

The interval in which $f(x)=\frac{x}{\log x}$ is increasing, is: