Question

Please help! Anything would be appreciated!

A manufacturer can produce a color pen at a cost of $3. The color pens have been selling for $5 per pen and at this price, consumers have been buying 4,000 pens per month. The manufacturer is planning to raise the price of the pens and estimates that for each $1 increase in the price, 400 fewer pens will be sold each month. At what price should the manufacturer sell the pens to maximize the profit? What is the profit?

Answer 1

$\bf \qquad \textit{vertex of a parabola}\\ \quad \\ y = {{ a}}x^2{{ +b}}x{{ +c}}\qquad \left(-\cfrac{{{ b}}}{2{{ a}}}\quad ,\quad {{ c}}-\cfrac{{{ b}}^2}{4{{ a}}}\right)$

so.. if you notice the graph below, that's when the profit is the highest, when the price is $\bf {{ c}}-\cfrac{{{ b}}^2}{4{{ a}}}$ and you'd be selling $\bf -\cfrac{{{ b}}}{2{{ a}}}$ pens