Help with question 3 and 6 please. Thank you! (1) A game console manufacturer determines that in order to sell x units, the price per oneunit (in dollars) must decrease by the demand function given by p(x) = 500- 0.1r,where p is measured in dollars/device. The manufacturer also determines that the costdepends on the volume of production and includes a fixed part, 100,000 (dollars), and avariable part, 100x (dollars), which gives us the cost function C(x) = 100000 + 100x.(a) What price per unit must be charged to get the maximum profit? How many gameconsoles must be sold to achieve the maximum profit? (Hint: Profit is determinedby subtracting costs from revenue, and you need to determine the revenue function.)(b) Find the marginal profit at a = 100 units and interpret the meaning of the solutionin context (Hint: The derivative of a function is called the marginal function.)(2) The demand function for a certain product is linear and defined by the equationp(x) = 10 – 5, where x is the total output. Find the level of production at which thecompany has the maximum revenue.2(3) A diesel generator burns fuel at the rate of G(x) = (338 + 2x) liters per hour whenproducing a thousand kilowatt hours of electricity. Suppose that fuel costs 60 cents aliter. Find the value of r that leads to minimum total cost if the generator is operatedfor 32 hours.12(4) To produce r units of some product a company spends C(x) = ax? + bx dollars, wherea and b are positive real numbers. The product is sold at price p dollars per unit.Determine the sales volume at which profit reaches its maximum.(5) The production cost per a period of time is given by the quadratic functionC(r) = a + bx², where a and b are some positive real numbers and x represents thenumber of units. Find the minimal average cost. (Hint: The average cost is the totalcost divided by the number of units produced.)%3D(6) A manufacturing company needs to build a box whose base length is 6 times the basewidth and they must ensure that the box will enclose 20 in of volume. The cost of thematerial of the sides is $3/in² and the cost of the top and bottom is $15/in?. Determinethe dimensions of the box that will minimize the cost.(7) CHALLENGE PROBLEM: A company sells wholesale lots of goods at a price of $20per thousand for orders of 10,000 goods or less, with the charge per thousand decreasedby 50 cents for each thousand above 10,000 goods. Determine the order size at which thecompany has the maximum revenue. (Hint: The revenue function is a piecewise-definedfunctionLet the independent variablebe the order size in thousands of goods....docW-Review for the f....docxMacBook Pro2.

For relative maximum or minimum To maximize or minimize a function f(x) 1st we need to find its critical points For critical points we need to put differentiation of f(x) equal to zero i.e. f '(x) = 0 Then check second derivative of f(x) i.e. f ''(x) is positive at critical numbers ( f ''(x) >0 ) then that gives minimum value of f(x) and if f ''(x) is negative ( f ''(x) <0 ) then that give maximum value of f(x) Note : If there is only one critical point then no need to check 2nd derivative