1. Find two numbers whose difference is 150 and whose product is a minimum. 2. 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