Problem 1: Marginal cost and average costs 1. Explainin words why the marginal cost curve must cross the average cost curve al the minimum of the average cost. MNow consider the following production function: / .2 S, K) = (VE+ VK] The quantity of capital in the short run is K = 9, and the input prices are w = 2 for labor and = 1 for capital. 2. Derive the short-run average cost, marginal cost, and average variable cost functions. (Hink: These functions are only defined for = 9, because if L = 0, the firm can still produce 2 f(0,9) = (\\5) =9 in the short run.) 3. Check that the short-run average cost reaches its minimum for = 'f1'_ (Hint: Show that the derivative of the SRAC function is equal to zero at = %.} What is the minimum value of the shorl-run average cost? 4. Verify analytically that at g = &, the SRMC and the SRAC intersect. (Hint: Show that the two functions have the same value at g = 81) 5. At what point do the short-run MC and the short-run AVC curves intersect? Comment. 6. Using the graphing software of your choice, plot the short-run marginal cost curve, the short run average cost curve, and the short-run average variable cost curve. Be sure to choose the output range correctly. In particular, your functions should only be defined for g = 9