The profit function for a certain commodity is P(x) = 180x − x2 − 1000. Find the level of production that yields maximum profit, and find the maximum profit.

x= 90 units, maximum profit =$7100

Explanation: p(x) = 180x - x² - 1000

By taking the first derivative of the profit function and equating the resulting function to zero and solve the resulting equation, we have the stationary point of the function which is the production level that yields maximum profit.

d{p(x)}/dx = 180 - 2x.

For stationary point, d{p(x)}/dx = 0

Hence 180 - 2x = 0

2x = 180, x = 90 units.

This implies that 90 units must be produced to attain maximum profit.

To get our maximum profit, we will now substitute the level at which we have maximum profit (x=90 units) into the profit function.

P(x) max = 180(90) - 90² - 1000

P(x) max = 16200 - 8100 - 1000

P(x) max = $7100

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