Exercise 10(h)

The profit function $\mathrm{P}\left(x\right)$ of a company selling $x$ items per day is given by
$P\left(x\right)=(150-x) x-1000$. Find the number of items that the company should manufacture to get maximum profit. Also find the maximum profit.

Find the absolute maximum and absolute minimum of $f\left(x\right)=8x^3+81x^2-42x-8$ on $\left[-8,2\right]$

Find two positive integers whose sum is $16$ and the sum of whose squares is minimum.

Find two positive integers $x$ and $y$ such that $x+y=60$ and $x{y}^3$ is maximum.

From a rectangular sheet of dimensions $30 \mathrm{cm}\times 80 \mathrm{cm}$, four equal squares of side $x \mathrm{cm}$, are removed at the corners, and the sides are then turned up so as to form an open rectangular box. Find the value of $x$, so that the volume of the box is the greatest.

A window is in the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is $20 \mathrm{ft}$, find the maximum area.

If the curved surface of right circular cylinder inscribed in a sphere of radius $R$ is maximum, show that the height of the cylinder is $\sqrt{2}R$.

A wire of length $l$ is cut into two parts which are bead respectively in the form of a square and a circle. What are the lengths of the pieces of the wire respectively so that the sum of the areas is the least.