SOLUTION: The profit function for a business is given by the equation P(x)=-4x^2+16x-7, where x is the number sold, in thousands, and P(x) is dollars in thousands. Find the maximum profit an

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Question 123462: The profit function for a business is given by the equation P(x)=-4x^2+16x-7, where x is the number sold, in thousands, and P(x) is dollars in thousands. Find the maximum profit and how many items must be sold to achieve it.
Answer by solver91311(24713) About Me  (Show Source):
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Maximize P%28x%29=-4x%5E2%2B16x-7

Two ways to solve this.

Without calculus:

This function describes a convex down parabola, so the maximum point is at the vertex. The x-coordinate of the vertex of parabola f%28x%29=ax%5E2%2Bbx%2Bc is at x=-b%2F2a

x%5Bmax%5D=-16%2F2%28-4%29=2

The value of the function at x=2 is the maximum profit.

f%282%29=-4%282%29%5E2%2B16%282%29-7=-16%2B32-y=9

So the profit is a maximum $9,000 when 2 units are sold.

Calculus:
A local minimum or maximum of a function is at the point where the first derivitive is equal to zero

dy%2Fdx=-8x%2B16
-8x%2B16=0
-8x=-16
x=2

If the second derivitive evaluated at this value is < 0, the point is a maximum.

d%5E2%28y%29%2Fdx%5E2=-8

Therefore the point is a maximum.

Find the function value at the maximum point just like before.