The function f(x) = –(x – 20)(x – 100) represents a company's monthly profit as a function of x, the number of purchase orders received. which number of purchase orders will generate the greatest profit?

The number of purchase orders to generate maximum profit is 60.

Step-by-step explanation:

We need to find the value of x which gives a maximum value of f(x).

f(x) = -(x^2 - 120x + 2000)

Converting to vertex from

= - [ (x - 60)^2 - 3600)  - 2000

= -- (x - 60)^2  + 3600 - 2000

= -(x - 60)^2 + 1600.

So the maximum value of f(x) is 1600 when x = 60.

When the number of purchase orders is 60, the monthly profit will be at its maximum possible for the month.

Step-by-step explanation:

The graph of this function is that of a parabola that opens down.  We know that because of the " - " sign in front of (x – 20)(x – 100).  We need to find the vertex of this parabola, as the y-coordinate tells us the max profit and the x-coordinate tells us at which number of purchase orders the profit is at a max.

Multiplying out the given function, we get:

f(x) = -(x – 20)(x – 100)

     = -(x^2 - 120x + 2000).  Here a = -1, b = 120 and c = -2000.

The x-coordinate of the vertex is

x = -b/[2a], which here is

x = -120 / [2*(-1)], or   x = 60.

The profit is the value of f(x) at x = 60.

When the number of purchase orders is 60, the monthly profit will be at its maximum possible for the month.

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step-by-step explanation: