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pletely correct. 1. A production function is given by f(K. L) - L/2+VK. Given this form. MP - 1/2 and MPk - JR. (a) Are there constant returns to scale, decreasing returns to scale, or increasing returns to scale? Explain why (b) Is the marginal product of labor constant, increasing, or decreasing (for fixed K)? (c) In the short run, capital is fixed at K .= 4 while labor is variable. On the same graph, draw the short run marginal product and average product curves. 2. A production function is f(L, K) - (L- + K"), where a >. ( and 6 2 0. For what values of a and b are there constant returns to scale? Increasing returns to scale? 3. Firm A has a production function fA(L, K) - min (21, I K ). Carefully graph the isoquants associated with production levels of Q = 10 and Q = 20. Does this exhibit constant, increasing, or decreasing returns to scale? 4. A firm has the production function /(L. K) = VL + A?. Given this, MA. and MPx 2K (a) For each of L and K, how does the marginal product of the factor change as the amount of the factor changes? (b) Show that this function does not satisfy the definition of either increasing, constant, or decreasing returns to scale. (Hint: Show that there are some values of I, and & for which this will show increasing or decreasing returns to scale.)Question 1 [25 marks]. (a) Let f : (a, b) - R be a real valued function. State the definition for f to be differentiable at a point x E (a, b). Geometrically, what does it mean for f to be differentiable at x? [5] (b) Consider the following function, g : IR - IR, given by 8 (x) = 12. Using the definition of derivative, compute the derivative of g. [5] (c) Consider the following function, h : IR - R, h(x) = [x). Show that h is not differentiable at x = 0. [5] (d) Let f : (a, b) - R be a differentiable function. Prove that f is continuous on (a, b). If f is continuous on (a, b), is it differentiable on (a, b)? (Justify your answer with a proof or counterexample.) [5] (e) Consider a differentiable function f : R = R which satisfies Using the equation above, show that f(x) Kexp(v) for some fixed number KER2. (a) Define, and carefully explain in your own words, what it means for [5] a function f : R - R to be differentiable at a point Xo E . (b) Plot the function f : R - R in a graph, where (10] f (x ) = |x -1| if x 5 2. if x > 2. (Recall |z) = z if z 2 0 and |z| = -z if z