Answer:
C
Step-by-step explanation:
The profit (in thousands of dollars) of a company is given by the function:
[tex]\displaystyle P(x) = -8x^2+32x+14[/tex]
And we want to find the maximum profit of the company.
Since the function is a quadratic with a negative leading coefficient, the maximum profit will occur at its vertex. Recall that the vertex of a quadratic is given by:
[tex]\displaystyle \text{Vertex} = \left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)[/tex]
Find the x-coordinate of the vertex. In this case, a = -8, b = 32, and c = 14. Hence:
[tex]\displaystyle x=-\frac{(32)}{2(-8)}=\frac{32}{16}=2[/tex]
To find the maximum profit, substitute this value back into the function. Hence:
[tex]\displaystyle P(2) = -8(2)^2+32(2) + 14 = 46[/tex]
Therefore, the maximum profit of the company is 46 thousand dollars.
Our answer is C.
