Q.A firm's cost function is $C(x) = 50x + 2000$ and its revenue function is $R(x) = 80x$, where $x$ is the number of units produced and sold. Find the profit function $P(x)$ and the break-even quantity.

Profit is revenue minus cost, and the break-even quantity is the output level at which profit is exactly zero.

The profit function is defined as revenue minus cost:

So $P(x) = 30x - 2000$ — a linear function of $x$, since both $R(x)$ and $C(x)$ are linear.

Break-even quantity: the break-even point is where profit is exactly zero, i.e. $P(x)=0$:

Since a fraction of a unit cannot actually be produced, the firm needs to sell at least $67$ whole units to move from a loss into a genuine profit; at exactly $66\tfrac23$ units the firm's revenue and cost are equal on paper. …

Students sometimes subtract the terms in the wrong order (computing $C(x)-R(x)$ instead of $R(x)-C(x)$), which flips the sign of the profit f …