Hw help please for Calculus 1

Question

If C(x) = 18,000 + 400x − 3.8x² + 0.004x³  is the cost function and p(x) = 4,000 − 8x is the demand function, find the production level (in units) that will maximize profit. (Hint: If the profit is maximized, then the marginal revenue equals the marginal cost.)

Answer 1

Revenue = xp(x) = 4,000x -8x^2

Marginal revenue is the derivative = R'(x) = 4000 - 16x

Marginal cost is the derivative of C(x) = C'(x) = 400 -7.6x+ .012x^2

set MR = MC and solve for x to get the production level that maximizes profit

R'(x) = C'(x)

simplifies to

x^2 +700x -300000 = 0

use the quadratic formula

x = 300, ignore the negative solution