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5. Consider a market for used cars, where there are 300 lemons and 600 peaches. Buyers value peaches at $12,000 and lemons at $6,000. (a) If all cars (lemons and peaches) clear the market, what is a condition on the seller's reservation price (willingness-to-sell) for peaches? (b) If sellers value peaches at $1 1,000 and lemons at $4,500, how many peaches sell? (c) Now suppose there are 300 grapefruits on the market (in addition to the 300 lemons and 600 peaches). Buyers value grapefruits at $8,000 and sellers are willing to sell them at $9,000. What types of cars are sold in the market? (d) Given your answer to part c), what is the buyer's new willingness-to-pay for a used car? (e) What types of cars are sold in the market?Question 1 (5 points) A1. A firm has a production function given by Q - MIN {L ; 5K). Over time, the production function evolves to become Q - MIN (2L ; 10K). Show that this represents technological progress. Add a File Rocard Augho Record Videoiton of ( mx), for all integers m. Show -m (*) = (1 + cos x)/2. Then find _ fam+1 (2).] 3(a) Let S be an arbitrary subset of R"; let Xo E S. We say that the fou. tion f : S - R is differentiable at xo, of class C", provided ( is a C" function g : U - R defined in a neighborhood U of 2. R", such that g agrees with f on the set Un S. In this case, that if o : R" - R is a C' function whose support lies in Up the function S $(x)g(x) for x E U, h(x) = for x @Support $, is well-defined and of class C' on R". (b) Prove the following: Theorem. If f : S - R and f is differentiable of class C at each point xo of S, then f may be extended to a C" functin h : A - R that is defined on an open set A of R" containing s [Hint: Cover S by appropriately chosen neighborhoods, let A their union, and take a Coo partition of unity on A dominated by this collection of neighborhoods.]ull workings required Lot 1: 842 - R be a differentiable function and let " C R. be a curve in R"2 described by the cartesian equation fly).20 Let(g.b) . be a point that lies on the curve ' cand assume that the partial derivatives of f evaluated at (a,b) satisfy : fa(a, b) 4 0 and fy(a. b) # 0. Also assume that there exists an expression y=g(x) that solves the equation fly)=0 for y in terms of x in a neighbourhood of the point (n h). This means that; rial = b And fix, g(x)) -0 Is an Identity in x in a neighbourhood of a al Show that ; and show each step of your derivation b) Let be the line tangent to the curve "at the point (a.b) Write down in terms of () a vector to the CR at the point (a.b) How is vector related to the gradient vector . . GR- explain your answer dJ Find the cartesian equation for the line / C R- Find a cartesian equation that describes the tangent plane to the graph of . on the pome ( pub in 1) Explain why is not a horizontal planeA firm has a production function given by Q = MIN {L ; 5K). Over time, the production function evolves to become Q = MIN (2L ; 10K). Show that this represents technological progress