Which of the following statements are true?
(A) The break-even point is the level of production where the revenue from sales is equal to the cost of production and marketing.
(B) If R is the total revenue collected by a company when it sells \( x \) units of a product at a price of p per unit, then R is given by R = \( px \).
(C) Let \( C=C(x) \) be the total cost of producing and marketing \( x \) units of a product, then marginal cost, \( MC=\displaystyle\frac{dC}{dx} \).
(D) If \( AC \) is the average cost, then \( AC=\displaystyle\frac{x}{C} \).
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A), (B) and (C) only
Let x units be produced and sold at price p per unit, with total cost C(x). Revenue is price × quantity, marginal cost is the rate of change of total cost, and average cost is the cost per unit.
So R = px, \( MC=\frac{dC}{dx} \), \( AC=\frac{C}{x} \), and at break-even R = C.
Checking the statements
(A) The break-even point is the level of production where the revenue from sales is equal to the cost of production and marketing. — Correct. At break-even R(x) = C(x), so there is neither profit nor loss.
(B) If R is the total revenue collected by a company when it sells \( x \) units of a product at a price of p per unit, then R is given by R = \( px \). — Correct. Revenue = price per unit × number of units.
(C) Let \( C=C(x) \) be the total cost of producing and marketing \( x \) units of a product, then marginal cost, \( MC=\displaystyle\frac{dC}{dx} \). — Correct. This is the definition of marginal cost.
(D) If \( AC \) is the average cost, then \( AC=\displaystyle\frac{x}{C} \). — Incorrect. Average cost is \( \frac{C}{x} \); the given expression is upside down.