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# Practice on Maximum and Minimum, Profit maximization

10 practice questions on Maximum and Minimum values

(2) Find the absolute maximum value and the absolute minimum value, if any, of the following function. (If an absolute maximum/minimum does not exist, enter NONE in that blank.) g(x) = -x^2 + 2x + 10
(6) Lynbrook West, an apartment complex, has 100 two-bedroom units. The monthly profit (in dollars) realized from renting out x apartments is given by the following function ...
(10) Phillip, the proprietor of a vineyard, estimates that the first 9300 bottles of wine produced this season will fetch a profit of \$3 per bottle. However, the profit from each bottle beyond 9300 drops by \$0.0003 for each additional bottle sold ...

All questions are in the attachment.

#### Solution Preview

The solution file is attached.

(1) (3, 1) (Absolute maximum)
(0, -3) (Absolute minimum)

(2) Find the absolute maximum value and the absolute minimum value, if any, of the following function. (If an absolute maximum/minimum does not exist, enter NONE in that blank.)

Since the leading coefficient is negative, g(x) has only a maximum and no minimum
g(x) = -(x^2 - 2x) + 10 = -(x^2 - 2x + 1) + 10 + 1 = -(x - 1)^2 + 11
The vertex is at (1, 11)

NONE (Absolute minimum)
11 (Absolute maximum)

(3) Find the absolute maximum value and the absolute minimum value, if any, of the following function. (If an absolute maximum/minimum does not exist, enter NONE in that blank.)

f'(x) = [(16 + x^2) - x(2x)]/(16 + x^2)^2 = (16 - x^2)/(16 + x^2)^2
f'(x) = 0  x = {-4, 4}
f"(x) = [(16 + x^2)^2 * (-2x) - (16 - x^2) * 2(16 + x^2) * 2x]/(16 + x^2)^4
f"(-4) > 0 and f"(4) < 0
f(x) has a minimum at x = -4 and a maximum at x = 4
The minimum value is f(-4) = -0.125 and the maximum value is f(4) = 0.125
-0.125 (Absolute minimum)
0.125 (Absolute maximum)

(4) Find the absolute maximum value and the absolute minimum value, if any, of the following function. (If an absolute maximum/minimum does not exist, enter NONE in that blank.)

Since the leading coefficient is negative, f(x) has only a ...

#### Solution Summary

The expert examines practices on maximum and minimum profit maximization.

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