Maharashtra Board Solutions for Chapter: Applications of Derivatives, Exercise 5: MISCELLANEOUS EXERCISE 2

Show that of all rectangles inscribed in a given circle, the square has the maximum area.

Show that a closed right circular cyclinder of given surface area has maximum volume if its height equals the diameter of its base.

A window is in the form of a rectangle surmounted by a semicircle. If the perimeter be $30 \mathrm{m}$, find the dimensions so that the greatest possible amount of light may be admitted.

Show that the height of a right circular cylinder of greatest volume that can be inscribed in a right circular cone is one-third of that of the cone.

A wire of length $l$ is cut in to two parts. One part is bent into a circle and the other into a square. Show that the sum of the areas of the circle and the square is least, if the radius of the circle is half the side of the square.

A rectangular Sheet of paper of fixed perimeter with the sides having their length in the ratio $8:15$ converted in to an open rectangular box by folding after removing the squares of equal area from all corners. If the total area of the removed squares is $100$, the resulting box has maximum valume. Find the lengths of the rectangular sheet of paper.

Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius $R$ is $\frac{2\mathrm{R}}{\sqrt{3}}$. Also find the maximum volume.

Find the maximum and minimum values of the function $f\left(x\right)={\cos }^{2}x+\sin x$.