Rolle's theorem holds for monic quadratic polynomial $f(x)$ on the interval $[\alpha, \alpha + 3]$ where $f(\alpha) = 0$. Similarly, $g(x) = f(x) + 2$ also follows Rolle's theorem in the interval $[\beta, 3]$ where $g(3) = 0$, such that the value of $c$ is the same for both $f(x)$ and $g(x)$. Then the value of $(f \circ g)(\alpha)$ is...

If the rate of increase of surface area of a spherical balloon is $5\,\text{cm}^2/\text{sec}$ and rate of increase of volume of a spherical balloon is $10\,\text{cm}^3/\text{sec}$, then the radius of the balloon at that time is...

A tank with a rectangular base and rectangular sides, open at the top is made. Depth of the tank is $4$ m and its volume is $36$ cubic meters. For making a tank cost of base material used is Rs. $100$ per sq. meter and that of sides is Rs. $50$ per sq. meter. Then minimum cost of tank is ..............

A particle is fired straight up from the ground. Its height in feet after $t$ second is given by $s(t) = 128t - 16t^2$. The velocity of the particle when it hits the ground is...