Find the maximum and minimum values of x + sin x on [ , π ]. . Find two numbers whose sum is and whose product is as large as possible. .
Find two positive numbers x and y such that x + y = and xy is maximum. . Find two positive numbers x and y such that their sum is and the product x y is a maximum. .
Find two positive numbers whose sum is and the sum of whose cubes is minimum. . A square piece of tin of side cm is to be made into a box without top, by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box is the maximum possible.
. A rectangular sheet of tin cm by cm is to be made into a box without top, by cutting off square from each corner and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is maximum ? .
Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area. . Show that the right circular cylinder of given surface and maximum volume is such that its height is equal to the diameter of the base. .
Of all the closed cylindrical cans (right circular), of a given volume of cubic centimetres, find the dimensions of the can which has the minimum surface area? . A wire of length m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle.
What should be the length of the two pieces so that the combined area of the square and the circle is minimum? . Prove that the volume of the largest cone that can be inscribed in a sphere of