(8a + 3) + (a + 2)
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Answer:
x=12 maximizes the profit function.
Step-by-step explanation:
We are given that the profit is for
To find x that maximizes P, we will find the derivative of P(x) and find x such that P'(x) =0. Recall that the derivative of a function of the form is
, and that the derivative of a constant is zero. Then, by using the properties of derivatives, we get (the details of the calculation is omitted).
.
We want to solve . By dividing the equation by -3, we get
So we have that x=12 and x=-2 are solutions. In this case, we are only considering x greater than o equals 0. So, we take x=12.
We will check that x=12 is a maximum of P.
To do so, we will use the second derivative criteria, which is as follows. Given a function f whose first and second derivative exist, a point x is a maximum if . In our case,
Note that . So x=12 is a maximum of P.