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Determining the profit-maximizing quantity of production
| 2 || The derivative of the profit function with respect to <math>Q</math> is <math>MR - MC</math> where <math>MR</math> is the marginal revenue (defined as the derivative of total revenue with respect to quantity, using price as a function of quantity via the market demand curve) and <math>MC</math> is the marginal cost (dependent on <math>Q</math>). || <toggledisplay>By the [[calculus:differentiation is linear|linearity of differentiation]], the derivative of the profit function is <math>Q \mapsto \frac{d}{dQ}(QP) - \frac{d}{dQ}(VC)</math>. The derivatives are <math>MR</math> and <math>MC</math> respectively.</toggledisplay>
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| 3 || For those quantities of production where <math>MR > MC</math>, the profit function is increasing with respect to <math>Q</math>. For those quantities of production where <math>MR < MC</math>, then the profit function is decreasing with respect to <math>Q</math>. || See [[calculus:positive derivative implies increasing|positive derivative implies increasing]], [[calculus:negative derivative implies decreasing|negative derivative implies increasing]].
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| 4 || The local maxima for the profit function must occur at points where <math>MR = MC</math>. Further, under reasonable assumptions of the function not being heavily oscillatory, we must have <math>MR > MC</math> for <math>Q</math> slightly less than the point of local maximum and <math>MR < MC</math> for <math>Q</math> slightly more than the point of local maximum. In other words, the <math>MC</math> curve is overtaking the <math>MR</math> curve. || Follows from (3). We can also interpret this in terms of the [[calculus:first derivative test|first derivative test]] applied to the profit function as a function of <math>Q</math>.
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