What is the second (or second order) condition for finding the maximum point of a function? Second derivative (or rate of change of the first derivative) is negative Second derivative is positive OSecond derivative (or rate of change of the first derivative) is zero Second derivative (or rate of change of the first derivative) is at a minimum
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- Differentiate the profit function with respect to ?1 ??? ?2.Explain intuitively why we take the derivative of the function.The partial derivative of U with respect to M is V2 not V2M. Right?A rental car company has a special deal on one of the available rentals. Let C be the cost, in dollars, of the rental car as a function of the distance d, in miles, it is driven in one day. A. What would the value C(75) represent in this context? B. Is function Cincreasing or decreasing? What are the units of the slope in this situation? C. Identify any maximum or minimum values of the function. What do they represent in this situation? State any assumptions that you make. D. Would the graph of C have any intercepts? What would they represent in this situation? E. Write a rule for C(d), and sketch a graph of the function.
- Consider the following Cobb-Douglas production function. Q=10 K1/2L1/2a. Find the first, second, and cross partial derivatives. b. Determine their signs.c. What is the economic interpretation of the signs of these derivatives?Consider the following Cobb-Douglas production function.Q=10 K1/2L1/2a. Find the first, second, and cross partial derivatives.b. Determine their signs.c. What is the economic interpretation of the signs of these derivatives?The monthly demand function for a particular product is q=f(p)=2400-15p, where q is stated inunits and p is the price in dollars.i) Determine (a) the Revenue function, (b) concavity, (c) revenue at a price of $50, (d) atwhat price will the revenue be maximized, (e) graph the revenue function.ii) In general how we determine Maximum and minimum value of Quadratic function.What does concavity tell us about the function’s behaviour?
- Analysis of daily output of a factory shows that, on average, the number of units per hour y produced after t hours of production is y = 44t + 0.5t2 − t3, 0 ≤ t ≤ 6. (a) Find the critical values of this function. (Assume −∞ < t < ∞. Enter your answers as a comma-separated list.) t = (b) Which critical values make sense in this particular problem? (Enter your answers as a comma-separated list.) t = (c) For which values of t, for 0 ≤ t ≤ 6, is y increasing? (Enter your answer using interval notation.)Define the strict gradient series?3. First, do the following functions exhibit increasing, constant, or decreasing returns toscale? Why? Explain in detail. Second, calculate the marginal product of each factorand discuss what happens to the marginal product of each individual factor as that factoris increased and the other factor is held constant. a. q = 3L + 4Kb. q = (2L + 2K)1/3c. q = 2LK2d. q = L1/4K1/3e. q = 4L1/2 + 4K
- . (a-) Graph the function Y= 36/X , assuming that X and Y can take positive values only. Next, suppose that both variables can take negative values as well; how must the graph be modified to reflect this change in assumption?You are working to model your production function and you are interested in the equation of the line tangent to the production function at a specific level of production. You know that the equation of the tangent line will be first derivative of the production function. Given this information, Previously, a consultant had estimated that your production function could be estimated as f(x) = -1,000 + 26 hours - 0.2 hours^2 Given that you firm is planning to use 97 hours in the production process, what is the value of the equation of the tangent line at 97?(i) Find the rate of change of the function f(x) =x + 2/ 1 − 8x with respect to x when x = 1.(ii) The number of units Q of a particular commodity that will be produced with Kthousand dollars of capital expenditure is modeled by Q(K) = 500 K2/3Suppose that capital expenditure varies with time in such a way that t months from nowthere will be K(t) thousand dollars of capital expenditure, whereK(t) =2t4 + 3t + 149 / t + 2 (a) What will be the capital expenditure 3 months from now? How many units will be producedat this time?(b) At what rate will production be changing with respect to time 5 months from now?Will production be increasing or decreasing at this time?