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2014-09-07T05:45:46+05:30

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Price at quantity q sold is = P(q) = 100 - 2 q

Revenue = R(q) = P(q) * q = 100 q - 2 q²

Marginal revenue = MR(q) = \frac{d R(q)}{d q} = 100 - 4 q \\

MR(q) = 0 => 100 - 4q = 0  => q = 25

Price at q = 25 is = P(25) = 100 - 2 * 25 = 50 units
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If we do the sum with out using differentiation to find MR(q) then:

MR(q) = revenue from selling quantity (q+1) - revenue from selling quantity q

 = R(q+1) - R(q) = P(q+1) * (q+1) - P(q) * q \\ \\ = [100 - 2(q+1)] (q+1) - (100-2q) q \\ \\ = (98-2q)(q+1) - (100-2q)q \\ \\ = 98 q +98 - 2q^2 - 2q - 100q + 2 q^2 \\ \\ = 98 - 4q \\

Marginal revenue = 0 => 4q = 98  => q = 49/2 units

P(q = 49/2) = 100 - 2 * 49/2 = 51 units

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