The widgets should be sold for $22.08 for the company to make the maximum profit.
To maximize a function, y = f(x), we differentiate it with respect to x, to get y' = f'(x). Equate it to zero to get the points on inflections.
Differentiate y' = f'(x) again with respect to x, to get y'' = f''(x), and put in the points of inflections to check whether y'' is greater than or less than zero. If it is greater than zero, then we have a minimum, and if it is less than zero, then we have a maximum.
In the question, we are given a profit function y, with respect to the sale price of each widget x. We are asked to find the sale price that maximizes the profit.
The function given is:
y = -30x² + 1325x - 8569.
Differentiating this with respect to x, we get:
y' = -60x + 1325.
To find the point of inflection, we equate this to zero, to get:
0 = -60x+ 1325,
or, x = 1325/60 = 22.0833
Now, we differentiate, y' = -60x + 1325, with respect to x, to get:
y'' = -60 < 0, thus we have a maximum at x = 22.0833.
Thus, the widgets should be sold for $22.08 for the company to make the maximum profit.
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