Question

If P(x) = – 14 + 9x– x^2 represents the profit in selling x thousand Boombotix speakers, how manyshould be sold to maximize profit?

Answer 1

The maximum and minimum points of a function happens at the zeros of the first derivative. The power rule is

$\frac{d}{dx}(x^n)=nx^{n-1}$

Using this property in our function, we're going to have

$\begin{gathered} \frac{d}{dx}P(x)=\frac{d}{dx}(-14+9x-x^2) \\ =\frac{d}{dx}(-14x^0+9x^1-x^2) \\ =\frac{d}{dx}(-14x^0)+\frac{d}{dx}(9x^1)-\frac{d}{dx}(x^2) \\ =(0)\cdot(-14x^{0-1})+(1)\cdot(9x^{1-1})-(2)\cdot(x^{2-1}) \\ =0+9-2x \\ =-2x+9 \end{gathered}$

which is a linear function, with a zero at

$\begin{gathered} -2x+9=0 \\ -2x=-9 \\ 2x=9 \\ x=\frac{9}{2} \\ x=4.5 \end{gathered}$

x = 4.5, therefore, 4500 Boombotix speakers should be sold to maximize the profit.