Mathematics : Application of Derivatives

Manufacturer can sell x items at a price of rupees  The cost price of x items is Rs.  Find the number of items he should sell to earn maximum profit. 

A firm has found from experience that its profit as a function of x, the amount produced, is given by

Find the value of x that maximises the profit.

A beam of length l is supported at one end. If W is the uniform load per unit length, the bending moment M at a distance x from the end is given by  maximum value. Find the point on the beam at which the bending moment has th

A wire of length 36 cm is cut into two pieces. One of the pieces is turned in the form of a square and the other in the form of an equilateral triangle. Find the length of each piece so that the sum of the areas of the two be minimum.

A window consists of a semi-circle with a rectangle on its diameter. If the perimeter of the window is 30 metres, find the dimensions of the window in order that its area may be maximum.
Or
A window is in the form of a rectangle surmounted by a semi-circle. If the total perimeter of the window is 30 m. find the dimensions of the window so that maximum light is admitted.

A wire of length 28 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?