Kindly help me solve these problems

A firm uses inputs of labor, L, and capital, K, to produce its output, Q, according to the production function Q=f(X,L)-9LUSKVS. The firm is a price-taker in the input markets. Labor is paid an hourly wage of w = 12, and the price of capital is r =6. The firm sells its output at a price of p = 4 per unit. 1) Maximize the profit function to determine the optimum level of each input the firm should use. 2) Check second-order sufficient condition. Show your calculations.Question 4 (6 points) Consider the market for used cars with two types: "lemons" (cars with problems) and "peaches" (cars without problems). Assume that a seller knows whether a car is a lemon or a peach, but a buyer cannot tell until the car is purchased. Suppose that the reservation price of a peach is $6.090 to the buyer and $5:009 to the seller. The reservation price of a lemon is $2.900 to the buyer and $500 to the seller. a. (3 points) Suppose that there are an equal number of peaches and lemons for sale and that buyers know this to be the case. In equilibrium, what proportion of car sales are peaches? b. (3 points) Now assume that the probability of getting a lemon is variable. Suppose that Jorge starts a business to sell tised cars with warranties, In particular, he buys all used cars in the market at $5.000 each (the reservation price of a peach's seller), and sells them at $6.000, of which $1:000 is the cost of The warranty. If the car is a lemon, he refunds the purchase price (that includes the warranty ), and resells the lemon at $2.000 without a warranty. Let p be the fraction of used cars that are lemons, For what range of p is this business profitable for Jorge?5. Suppose N has mean 3 and variance {. Suppose )' has mean S and Variance 9. Let Z - 6X + 7Y. Suppose the covariance between X and ) is 2 (a) Compute the mean of Z [b) Compute the variance of Z (e) Compute the variance of 6X - TY (d) Com inmor between X and Z2. Suppose that X has mean fx and variance of, and suppose that Y has mean by and variance of. Suppose the correlation between X and Y is p. Let U = ax +b and V = cy + d where a, b, c, d are constants. (a) Find Covariance(U. V) in terms of a, b. c, d, ux. ox, ur. of, and p. (Note: some of these parameters might not appear in the answer.) (b) Find Corr(U. V) in terms of a, b. c, d, ux. ox. My, or, and p. (Note: some of these parameters might not appear in the answer.) (c) Suppose that W and V are two random variables with Var(W) = 4 Var(V) = 9, and Cov(W. V ) = 2. Find Var(-4W + 3V - 2). (d) Suppose that X and Y are random variables with cov(X, Y) = 3 Find cov(2X - 4, 3 - 5Y)