Question

A profit function is derived from the production cost and revenue function for a given item. The monthly profit function for a certain item is given by P(x)=−0.05x^2+500x−100,000, where P is in dollars and x is the number of units sold. How many units maximize the profit? FInd the maximum profit

Answer 1

Check the picture below.

$\bf \textit{ vertex of a vertical parabola, using coefficients}\\\\ \begin{array}{lccclll} P(x) = &{{ -0.05}}x^2&{{ +500}}x&{{ -100000}}\\ &\uparrow &\uparrow &\uparrow \\ &a&b&c \end{array}\qquad \left(-\cfrac{{{ b}}}{2{{ a}}}\quad ,\quad {{ c}}-\cfrac{{{ b}}^2}{4{{ a}}}\right)\\\\ -------------------------------\\\\ \textit{so }-\cfrac{b}{2a}\textit{ units maximize the profit, and}\\\\\\ c-\cfrac{b^2}{4a}\textit{ dollars is the maximum profit}$