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

`(d)/(dx)(x^n)=nx^(n-1)`

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

`\begin{gathered} (d)/(dx)P(x)=(d)/(dx)(-14+9x-x^2) \\ =(d)/(dx)(-14x^0+9x^1-x^2) \\ =(d)/(dx)(-14x^0)+(d)/(dx)(9x^1)-(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=(9)/(2) \\ x=4.5 \end{gathered}`

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