is to be constructed by removing equal squares from each corner of a metre by metre rectangular sheet of aluminium and folding up the sides. Find the volume of the largest such box. Solution Let x metre be the length of a side of the removed squares. Then, the height of the box is x , length is – x and breadth is – x (Fig .
). If V( x ) is the volume of the box, then Fig . V( x ) = x ( – x ) ( – x ) = x – x + x Therefore V ( ) ( )( ) V ( ) ′ ′′ Now V ′ ( x ) = gives , x = . But x ≠ (Why?) Thus, we have x = .
Now V ′′ = − Therefore, x = is the point of maxima, i.e., if we remove a square of side metre from each corner of the sheet and make a box from the remaining sheet, then the volume of the box such obtained will be the largest and it is given by V = m Example Manufacturer can sell x items at a price of rupees each. The cost price of x items is Rs . Find the number of items he should sell to earn maximum profit. Solution Let S( x ) be the selling price of x items and let C( x ) be the cost price of x items.
Then, we have S( x ) = and C( x ) = x + Thus, the profit function P( x ) is given by P( x ) = S( ) C( ) i.e. P( x ) = x − P ′ ( x ) = Now P ′