## Pierre de Fermat

#### Exercises

1. An optimization problem: A gardener has 120 ft of fencing to enclose two adjoining rectangular plots of land as in the diagram below.  What dimensions should the garden be to maximize the enclosed area?  [Hint: if x and y are the unknown dimensions, write equations that represent the total length of the fencing and the area of the garden; solve for y in the first and substitute into the second to obtain a quadratic function in x which you must maximize.]
1. Another optimization problem: A producer of novelty items can sell 1000 items per week at a price of \$5 per item, and guesses that for every \$.10 drop in the price, it can sell 100 more items weekly.  The company has fixed overhead costs of \$1050 associated with manufacture and distribution in addition to costs of \$1.10 per unit in materials.  How many units should it produce and sell to maximize its profit?  [Hint: revenue = price per item times the number of units produced (and sold), and profit = revenue minus cost.]
2. Find the equation of the tangent line to the parabola which is the graph of y = x2 + 2x at the point (2, 8) by computing a derivative.
3. At what points on the graph of the equation y = 2x3 + 3x2 - 6x + 1 is the tangent line horizontal?