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Largest profit

A profit that is a quadratic with a negative coefficient of x squared has a maximum. The maximum lies on the line of symmetry, where the derivative is zero. You find the x-value, insert it into the profit and check that the graph opens downwards. That is the production that gives the largest profit. The endpoints of a closed interval must also be checked if the task limits the number of units.

P(x) = -ax^2 + bx + c

profit, a > 0

x = \frac{b}{2a}

maximum

Symbols

$a,\ b,\ c$	coefficients
$P$	profit	NOK

Example

$P(x) = -x^2 + 40x - 100$ . $x = \dfrac{40}{2} = 20$ .

$P(20) = -20^2 + 40 \cdot 20 - 100 = 300$ NOK.

A negative coefficient of x squared gives a maximum, not a minimum.

Practise differentiation and optimisation for free →

← Marginal cost and unit cost · Independent events →

Part of Mathematics S1: Differentiation and optimisation.