The function P (l) = -4l^2 + 32l - 52 gives the profit in thousands of producing l units of lip gloss. What is the maximum profit that can be made?
To find the maximum profit, we need to find the vertex of the parabolic function P(l) = -4l^2 + 32l - 52.
The vertex of a parabola in the form y = ax^2 + bx + c is given by the formula x = -b/(2a).
In this case, a = -4 and b = 32. Plug these values into the formula:
l = -32/(2*(-4)) = -32/-8 = 4
So, the maximum profit is obtained when l = 4. Now, plug l = 4 back into the profit function to find the maximum profit:
P(4) = -4(4)^2 + 32(4) - 52
P(4) = -4(16) + 128 - 52
P(4) = -64 + 128 - 52
P(4) = 64 - 52
P(4) = $12,000
Therefore, the maximum profit that can be made is $12,000.