A company has calculated that the daily cost (in euro) to produce x items is given by the production cost function C(x) = 5x² + 750x + 3000 - Leaving Cert Mathematics - Question Question 1 - 2013

Profit is defined as total income minus total costs:

$Profit = R(20) - C(20)$

Substituting the values: $Profit = 24000 - 24000 = €0$

The profit function P(x) can be expressed as:

$P(x) = R(x) - C(x) = 1200x - (5x^2 + 750x + 3000)$

Thus simplifying:

$P(x) = -5x^2 + 450x - 3000$

To maximize profit, we take the derivative of the profit function and set it to zero:

$P'(x) = -10x + 450 = 0$

Solving for x gives: $x = 45$

Substituting x = 45 into the profit function:

$P(45) = -5(45)^2 + 450(45) - 3000$

Calculating: $P(45) = -10125 + 20250 - 3000 = €7,125$

Setting the cost function equal to €11,000:

$5x^2 + 750x + 3000 = 11000$

This simplifies to:

$5x^2 + 750x - 8000 = 0$

Using the quadratic formula gives the valid solution for x:

$x=10$