Q.For a certain product, the revenue function is $R(x) = 60x$ and the cost function is $C(x) = 20x + 8000$, where $x$ is the number of units sold.

Profit is revenue minus cost; break-even is where this difference equals zero. …

1. Profit function.

   $P(x) = R(x) - C(x) = 60x - (20x+8000) = 40x - 8000$

2. Break-even quantity. Break-even occurs where profit is zero:

   $40x - 8000 = 0 \Rightarrow 40x = 8000 \Rightarrow x = 200 \text{ units}$

   Independent cross-check: at $x=200$, $R(200)=60(200)=12{,}000$ and $C(200)=20(200)+8000=4000+8000=12{,}000$ — revenue exactly equals cost, confirming break-even. …

Writing profit as C(x) − R(x) instead of R(x) − C(x) (which would give the wrong sign throughout). Also, forgetting to bracket the whole cost function when subtra …