All courses › Mathematics S2 › Minimum unit cost

Minimum unit cost

The unit cost divides the whole cost by the number of units. With a quadratic term, a linear term and a fixed term it is an expression with x, a constant and a term with x in the denominator. The derivative is zero when the quadratic contribution and the fixed contribution weigh the same, that is when x is the square root of the fixed term divided by the coefficient of x squared. Insert it to get the amount.

A(x) = ax + b + \frac{c}{x}

unit cost

x = \sqrt{\frac{c}{a}}

the minimum

Symbols

$a,\ b,\ c$	coefficients in C
$A$	unit cost	NOK

Example

$C(x) = 2x^2 + 50x + 200$ . The minimum is at $x = \sqrt{200/2} = 10$ .

$A(10) = 2 \cdot 10 + 50 + \dfrac{200}{10} = 90$ NOK.

x is positive in a production, so we take the positive root.

Practise cost, revenue and profit for free →

← Optimal production · Marginal cost →

Part of Mathematics S2: Cost, revenue and profit.