Optimizing a Profit Function?

📌 CONCEPT: To maximize the profit function P(x) = 200x - 0.2x^2, we need to find the production level x that results in the maximum value of P(x) within the given constraint that x ≤ 1000.

📐 RULE / FORMULA: We can use the concept of completing the square to rewrite the profit function in the form P(x) = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. The maximum value of the profit function occurs at the vertex of the parabola.

💡 WORKED EXAMPLE: Suppose we want to find the maximum profit when producing a maximum of 1000 units. We can rewrite the profit function P(x) = 200x - 0.2x^2 as P(x) = -0.2(x^2 - 1000x) = -0.2(x - 500)^2 + 50000. From this form, we can see that the maximum profit occurs when x = 500 units.

⚠️ COMMON MISTAKE: Students often get trapped into using the first derivative test to find the maximum value of the profit function. However, the first derivative test may not give the correct result because the function may have a local maximum. Therefore, we should use the concept of completing the square to rewrite the profit function in the form P(x) = a(x - h)^2 + k and find the vertex of the parabola, which gives the maximum value of the profit function.