Calculus Grade 12 University

The production level that maximizes profit can be found by setting the revenue function equal to the cost function and solving for x. This yields an x-value of 171,428.093. This indicates that producing 171,428 felt-tipped pens produces the maximum profit from the given cost and revenue functions. To explain this mathematically, the profit, P(x), can be expressed as: P(x) = R(x) - C(x) = 0.68x - 0.00001x^2 - [48 000 + 0.28x + 0.00001x^2] = 0.40x - 48 000 We want to maximize this profit, so we take the derivative of the profit (with respect to x) to find the x-value at which the profit is at its maximum: P'(x) = 0.40 = 0 x = 171,428.093 Hence, producing 171,428 felt-tipped pens produces the maximum profit from the given cost and revenue functions.