Business Mathematics and Basic Statistics · Ch 10 — Limits and Derivatives

Application: Maxima and Minima for Cost, Demand and Marginal-Cost Functions

Differentiation has a direct business use: finding the output level at which a cost function is smallest, or a revenue/profit function is largest. At any point where a smooth function reaches a peak (maximum) or a trough (minimum), its graph is momentarily flat — so the derivative at that point is exactly $0$ .

Note

Method: Locating and Classifying a Maximum or Minimum

Find the critical point(s): solve $\dfrac{dy}{dx}=0$ for $x$ . Each solution is a candidate output level for a maximum or a minimum.
Classify using the second derivative: compute $\dfrac{d^2y}{dx^2}$ $\frac{d ^{2} y}{d x ^{2}}$ (differentiate $\dfrac{dy}{dx}$ $\frac{d y}{d x}$ once more, using the same §7 formulae/chain rule) and evaluate it at the critical point.
- If $\dfrac{d^2y}{dx^2} < 0$ at that point, $y$ has a maximum there.
- If $\dfrac{d^2y}{dx^2} > 0$ at that point, $y$ has a minimum there.

In business terms:

Given a total cost function $C(x)$ , the marginal cost function is $MC(x) = \dfrac{dC}{dx}$ — the extra cost of producing approximately one more unit near output level $x$ .
Given a demand function relating price $p$ to quantity $x$ (e.g. $p = a - bx$ ), the revenue function is $R(x) = p\cdot x$ , and revenue is maximized exactly where $\dfrac{dR}{dx}=0$ and $\dfrac{d^2R}{dx^2}<0$ .
Given a profit function $\Pi(x) = R(x) - C(x)$ , profit is maximized where $\dfrac{d\Pi}{dx}=0$ (equivalently, where marginal revenue equals marginal cost) and $\dfrac{d^2\Pi}{dx^2}<0$ . …

Definition 12Marginal cost / marginal revenue

$MC(x)=\dfrac{dC}{dx}$ and $MR(x)=\dfrac{dR}{dx}$ — the derivative of a total cost/revenue function, read as the approximate extra cost/revenue from one …

Definition 13Second-derivative test for maxima/minima

At a critical point where $\dfrac{dy}{dx}=0$ : if $\dfrac{d^2y}{dx^2}<0$ the point is a maximum; if $\dfrac{d^2y}{dx^2}>0$ t …

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