[babygirl200]

given P(x) = 1/2x^2 + 400x - 5000, find average profit, the minimum average profit and the x value

[ArcSine]

Your given function P(x) gives total profit (as a function of units x). Since average profit is just total profit divided by the number of units which generate such profit, your average profit function is



Replace that numerator by your algebraic expression for P(x), and you've got your average profit formula.

For the minimum, I suspect you're looking for "minimum profit" rather than "minimum average profit". Check the question to verify. If so, note that the total profit function P(x) is a convex (opening upward) parabola. You can find its minimum in either of two ways, depending on which you're more familiar with:

Algebraically, find the vertex of the parabola. The coordinates (x, y) of the vertex will give you, respectively, the number of units at which minimum profit is achieved, and the amount of such minimum profit.

Or, take the first derivative of the total profit function, and find the x-value for which the derivative is zero. As a quadratic function, total profit's derivative will have only one such point, and as a concave parabola, it'll represent the minimum profit.