If $x$ and $y$ are two positive integers such that $x + 2y = 10$ and $x^2 y^3$ is maximum,then $x^2 + 2y^3 =$

$f(x) = \int\limits_0^x t(t - 1)(t - 2) dt$ takes on its minimum value when:

The maximum area of a parallelogram inscribed in a triangle $ABC$ (given that $A$ is also one of the vertices of the parallelogram) is equal to (where symbols have their usual meaning in $\Delta ABC$):

Find the local maximum and local minimum values of the function given by $f(x) = x^{3} - 6x^{2} + 9x + 15$.

If $0 < x < \frac{\pi}{2}$,then the maximum area (in sq. units) of the triangle whose vertices are $(0,0)$,$(x, \cos x)$ and $(\sin^3 x, 0)$ is

Let $f(x)=(x+3)^2(x-2)^3$ for $x \in [-4,4]$. If $M$ and $m$ are the maximum and minimum values of $f$ respectively in $[-4,4]$,then the value of $M-m$ is: