Maximize profit - Math Central

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Question from Chris, a student:

A firm has the following total revenue and total cost function.
TR=100x-2x^2
TC=1/3x^3-5x^2+30x
Where x=output
Find the output level to minimize profit and the level of profit achieved at this output.

Hi Chris,

Profit $P$ is revenue minus cost so for your firm that's

\[P = \left( 100x-2x^2 \right) - \left( \frac13 x^3-5x^2+30x \right)\]

Simplify this expression use the derivative to determine the value of $x$ that maximizes the profit.

Penny

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