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Linear Programming Problems

A firm has th...

Question

A firm has the cost function $C = \frac{x^{3}}{3} - 7 x^{2} + 111 x + 50$ and demand function $x = 100 - p$ .
Formulate the total profit function $P$ in terms of $x$

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Solution

Cost function is $C (x) = \frac{x^{3}}{3} - 7 x^{2} + 111 x + 50$
Demand function is $x = 100 - p \Rightarrow p = 100 - x$
Revenue function, $R (x) = p . x = x (100 - x) = 100 x - x^{2}$
Profit function, $P (x) = R e v e n u e - C o s t$
$= R (x) - C (x)$
$= 100 x - x^{2} - \frac{x^{3}}{3} + 7 x^{2} - 111 x - 50$
$= - \frac{x^{3}}{3} + 6 x^{2} - 11 x - 50$

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