Divide  a number 15 into two parts such that the square of one multiplied with the cube of the other is a maximum.

Divide 15 into two parts such that the sum of squares is minimum.

How should we choose two numbers, each greater than or equal to -2 whose  sum is  so that the sum of the square of the first and cube of the second is the minimum?

The space s described in time t by a particle moving in a straight line is given by s = r⁵ – 40r + 30r² + 80t – 250. Find the minimum value of its acceleration.

Two sides of a triangle are given. Find the angle between them such that the area shall be maximum.

Show that of all the rectangles with a given perimeter, the square has the largest area.

287.

Find the area of the largest rectangle having the perimeter of 200 metres.

Show that among rectangles of given area, the square has the least perimeter.

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

Find the dimensions of the rectangle of greatest area that can be inscribed in a semi-circle of radius r.