Let the sum of the maximum and the minimum values of the function $f(x) = \frac{2x^2 - 3x + 8}{2x^2 + 3x + 8}$ be $\frac{m}{n}$,where $\gcd(m, n) = 1$. Then $m + n$ is equal to :

The local maximum value of $f(x) = x + \frac{1}{x}$ for $x < 0$ is equal to:

Let $f(x) = \begin{cases} |x| & \text{for } 0 < |x| \leqslant 2 \\ 1 & \text{for } x = 0 \end{cases}$. What is the nature of $f$ at $x = 0$?

The abscissa of the points,where the tangent to the curve $y=x^3-3x^2-9x+5$ is parallel to the $X$-axis,are

Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius $r$ is $\frac{4r}{3}$.

For a steamer,the consumption of petrol (per hour) varies as the cube of its speed (in $km/hr$). If the speed of the current is steady at $C \, km/hr$,then the most economical speed of the steamer going against the current will be ........... $C$.