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The profit function for a business is given by the equation P(x)=−4x2+16x−7, where x is the number of items sold, in thousands, and P(x) is the profit in thousands of dollars.  Calculate the maximum profit and the number of items that must be sold to achieve it.

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you can calculate the derivative of this function and find when it will be equal to zero since we know that such afunction as only one single tip for wich the slope is equal to zero. So finding for wich value of x the derivative of this function is zero, here, will give you the value of x for which you reach the maximum, so for how many item sold you'll reach it.

P'(x) = -8x + 16
P'(x) = 0    x = ?
0 = -8x +16
8x = 16
x = 2

P'(x) is the derivative of the finction you gave me and 2 is then the number of items sold in thousdans for wich you reach a maximum profit. Now to calculate that maximum profit, just substitute 2 for x in P(x).

P(2) = ?
P = -4(2^2)+ 16(2) - 7
P = -4(4) + 32 - 7
P = -16 + 32 - 7
P = 9

So the maximum profit would be 9,000 $ for 2,000 items sold.

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