Maxima and Minima (Profit Maximisation)

A central business application of differentiation is finding the level of output at which a quantity — most importantly, profit — is at its maximum (or, for cost, at its minimum). The GSEB Std 12 Commerce Maths and Statistics exam pattern places consistent weight on this maxima-minima business application, so the method below is worth mastering thoroughly.

First Derivative Test — Locating Critical (Stationary) Points

At a maximum or a minimum point of a smooth curve $y = f(x)$, the tangent to the curve is horizontal — its slope is zero. So the first step to locate a maximum or minimum is to solve:

Each solution of this equation is called a critical point (or stationary point). A critical point COULD be a maximum, a minimum, or neither (a point of inflection) — the first derivative test alone does not say which.

Second Derivative Test — Confirming Maximum or Minimum

To determine the NATURE of a critical point $x = x_0$ (found from $\frac{dy}{dx}=0$), examine the second derivative $\dfrac{d^2y}{dx^2}$ at that point:

• If $\dfrac{d^2y}{dx^2} < 0$ at $x_0$: the curve is concave downward there, so $x_0$ gives a maximum.
• If $\dfrac{d^2y}{dx^2} > 0$ at $x_0$: the curve is concave upward there, so $x_0$ gives a minimum.
• If $\dfrac{d^2y}{dx^2} = 0$: the test is inconclusive; further checking is needed (rare in routine commerce problems).

Profit Maximisation

Profit is $\pi(Q) = TR(Q) - TC(Q)$. To find the output level that maximises profit:

$\frac{d\pi}{dQ} = \frac{d(TR)}{dQ} - \frac{d(TC)}{dQ} = MR - MC = 0 \quad\Rightarrow\quad MR = MC$ …