Question

The function f(x) = -2(x - 10)(x -200) represents a company's monthly profit as a function of the number of items sold, x. What is the maximum monthly profit?

Answer 1

The company's maximum monthly profit is $18050

Explanation:

First we need to distribute everything in order to solve for the maximum

`f(x)=-2(x-10)(x-200)\\\\f(x)=-2(x^2-210x+2000)\\\\f(x)=-2x^2+420x-4000`

Now we can use the equation
`x=(-b)/(2a)` in order to find the x-value of the maximum

From our function, we know that a=-2 and b=420

We can plug in these values and solve for x to get

`x=(-420)/(-2(2)) \\\\x=(420)/(4) \\\\x=105`

Now we can plug our x-value into our function in order to find the maximum monthly profit

`f(x)=-2x^2+420x-4000\\\\f(105)=-2(105)^2+420(105)-4000\\\\f(105)=-2(11025)+44100-4000\\\\f(105)=-22050+44100-4000\\\\f(105)=18050`