Question

the average cost for a company to produce x items is x^2-10x+43 dollars. how many items should the company produce to minimize the average cost per item?

Answer 1

The cost equation is a parabolic graph, whose "lowest value" is at its vertex, it is up and goes down down down, reaches the U-turn and then goes up up up again.

$\bf \textit{ vertex of a vertical parabola, using coefficients}\\\\ \begin{array}{llll} y = &{{ 1}}x^2&{{ -10}}x&{{ +43}}\\ &\uparrow &\uparrow &\uparrow \\ &a&b&c \end{array}\qquad \left(-\cfrac{{{ b}}}{2{{ a}}}\quad ,\quad {{ c}}-\cfrac{{{ b}}^2}{4{{ a}}}\right) \\\\\\ \left( -\cfrac{(-10)}{2(1)}~~,~~43-\cfrac{(-10)^2}{4(1)} \right)\implies (\stackrel{\textit{items to produce}}{5}~~,~~\stackrel{\textit{production cost}}{18})$