Determine the level of output for which profit is maximized

Question

1. a. Test the function f(x) = 1/3x³ + 3x² - 27x + 10 for relative extrema.

b. Examine the function f(x) = x²ex - 2xe^x + e^x for turning points and classify as maximum, minimum or point of inflexion.

2. The total revenue of a firm is R(q) = 21q + 2q2 and its total cost

C(q) = -1/3q³ -2q² - 15q - 450. Where q is the firms output.

a. Is the rate of change of profit increasing or decreasing when the ouput level of the firm is q = 10?

b. Determine the level of output for which profit is maximized.

3. The demand curve of a firm is p = 4000 - 33q and its total cost C(q) = 2q³ - 3q² + 400q + 5000. Where q is the firm's output

a. Derive an expression R(q) for the firm's total revenue function.

b. Derive an expression 11(q) for the firm's profit function.

c. Is the rate of change of profit increasing or decreasing when the ouput level of the firm is q = 10?

d. Determine the level of output for which profit is maximized.