Calculus with economics

Let x be the number of thousands of units of an item produced. The revenue for selling x units is r(x) = 4√x and the cost of producing x units is c(x) = 2x² . Write an expression for the profit p(x).

• p(x) = 2x² - 4√x

• p(x) = 4√x - 2x²

• p(x) = 2x² + 4√x

• p(x) = 4√x + 2x²

Find the first derivative of the profit function!

• p'(x) = 2x^(-0.5) + 4x

• p'(x) = 4x - 2x^(-0.5)

• p'(x) = 2x^(0.5) - 4x

• p'(x) = 2x^(-0.5) - 4x

Find the second derivative of p(x).

• p''(x) = -x^(-3/2) - 4

• p''(x) = -x^(-3/2) + 4

• p''(x) = 4 - x^(-3/2)

• p''(x) = -x^(3/2) - 4

Hence find the number of units that should be produced in order to maximize the profit.

• 0.629

• 63

• 630

• 6290

What tells you that your answer is really the maximum of p(x)?

• it is the value that makes p'(x) zero

• it is the value that makes p''(x) zero

• it is a value for which p'(x) = 0 and p''(x) is positive

• it is a value for which p'(x) = 0 and p''(x) is negative