1) The cost per unit produced at a certain facility is represented by the function UC = 2x^2 - 10x + 50, where x is in thousands of units produced. For what value of x would unit cost be minimized (other than zero)? What is the minimum cost at this volume? Show that the value found is truly a minimum.

2) Advertising expenditures have been found to relate to profit approximately in accordance with the function P = x^3 - 100x^2 + 3125x, where x is the expenditure in thousands of dollars. What advertising expenditure would produce the maximum profit? What profit is expected at this expenditure? Show that the derived result is truly a maximum.

Because, cost per unit is function of number of units x, to find value of x for which cost is minimum, differentiate cost c with x:
dc/dx = 2*2*x - 10 + 0 = 4*x - 10

For maxima or minima,
dc/dx = 0
=> 4*x - 10 = 0
=> x = 10/4 = 2.5

To confirm maxima or minima, differentiate second time,
d^2(c)/dx^2 = (d/dx)(dc/dx) = 4*1 - 0
=> d^2(c)/dx^2 = 4

As, second derivative is +ve, means at x = 2.5, c has a minima.

Hence, value of x for which unit cost is minimum, is 2.5*1000 == 2500 units.

For given cost function, optimal size of production is estimated to minimize cost and the minimum cost is estimated. In 2nd problem, advertising cost function is given and optimal production size is estimated to maximize profit

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