1. Find two numbers whose difference is 150 and whose product is a minimum. Find two positive numbers with product 200 such that the sum of one number and twice the second number is as small as possible. 3. A rectangle has a perimeter of 100 cm. What length and width should it have so that its area is a maximum? 4. Show that a rectangle with a given area has a minimum perimeter when it is a square. 5. A box with a square base and open top must have a volume of 400 cm' . Find the dimensions of the box that minimizes the amount of material used. 6. A box with an open top is to be constructed from a square piece of cardboard that is 3 m wide, by cutting out a square from each from each of the four corners and bending up the sides. Find the largest volume that such a box can have. 7. A farmer wants to fence an area of 750 ooo m' in a rectangular field and divide it in half with a fence parallel to one of the sides of the rectangle. How can this be done so as to minimize the cost of the fence?8. Find the point on the line y = 5x + 4, that is closest to the origin. 9. Find the point on the parabola 2y = x that is closest to the point (- 4,1). 10. A can is to be made to hold a litre of oil. Find the radius of the can that will minimize the cost of the metal to make the can. ( 1L = 1000 cm') 11. A piece of wire 40 cm long is cut into two pieces. One piece is bent into the shape of a square and the other is bent into the shape of a circle. How should the wire be cut so that the total area enclosed is a) a maximum ? b) a minimum? 12. A rectangle is inscribed in a semicircle of radius 2 cm. Find the largest area of such a rectangle