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 ........... $C$.

If $f(x) = x^5 - 5x^4 + 5x^3 - 10$ has local maximum and minimum at $x = p$ and $x = q$,respectively,then $(p, q)$ equals

The displacement $x$ of a particle at time $t$ is given by $x = t^4 - kt^3$. If the velocity of the particle is maximum at $t = 2$,then $k = $ ..........

Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in $sq.m$) of the flowerbed is

Let a function $f(x)$ be continuous in an interval $[a, b]$. Let $\delta > 0$ be a very small real number. Let $c \in (a, b)$ be such that $f(c - \delta) < f(c)$ and $f(c + \delta) < f(c)$ for every $\delta > 0$. Let $(f(\alpha - \delta) - f(\alpha))(f(\alpha + \delta) - f(\alpha)) < 0$ for all $\alpha \in (a, b)$ and $\alpha \neq c$. Then:

Point $A$ lies on the curve $y = e^{-x^2}$ and has the coordinates $(x, e^{-x^2})$ where $x > 0$. Point $B$ has the coordinates $(x, 0)$. If $O$ is the origin,then the maximum area of the triangle $AOB$ is