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Find the marginal profit of a firm with a profit function:

$$P(q) = -192q + 88q^2 - 16q^3$$

I got

$$\frac{dP}{dq} = -192 + 176q - 48q^2$$

However, the solution reads

$$ \frac{dP}{dq}= -4(q-2)(q-4)(q-6)$$

Is this a typo, or am I missing something?

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closed as off-topic by Giskard, Herr K., ml0105, optimal control, Bayesian Feb 10 '17 at 17:30

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question does not meet the standards for homework questions as spelled out in the relevant meta posts. For more information, see our policy on homework question and the general FAQ." – Giskard, Herr K., ml0105, optimal control, Bayesian

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The marginal profit you calculate is correct.

We can rearrange the solution of the problem you are given. This is equivalent to

$$ \frac{dP}{dq} = 192 -176q + 48q^2 -4q^3$$

This derivative has as primitive function the following profit function:

$$P(q) = c + 192q -88q^2 + 16q^3 - q^4$$

where $c$ is a constant (e.g. $c=0$).

This is clearly different from the original profit function in your problem. In consequence, either the profit function or the solution is wrong.

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