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

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) (Show Source): You can put this solution on YOUR website!
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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 is at

The value of the function at is the maximum profit.

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

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

Therefore the point is a maximum.

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

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