A firm faces the demand $p = 50 - x$ (price in ₹ when $x$ units are sold) and total cost $C(x) = 100 + 20x$. Find the output that maximises profit, the maximum profit, and the price charged. Verify that MR = MC at this output.

Question

Accepted Answer

**Revenue.** $R(x) = p\,x = (50 - x)x = 50x - x^2$.

**Profit.**
$$\pi(x) = R(x) - C(x) = (50x - x^2) - (100 + 20x) = 30x - x^2 - 100.$$

**Maximise.** Differentiate and set to zero:
$$\pi'(x) = 30 - 2x = 0 \ \Rightarrow\ x = 15.$$
Second derivative: $\pi''(x) = -2 < 0$, so $x = 15$ gives a **maximum**.

**Maximum profit.**
$$\pi(15) = 30(15) - (15)^2 - 100 = 450 - 225 - 100 = 125.$$
So the maximum profit is ₹125.

**Price charged.** From the demand, $p = 50 - x = 50 - 15 = 35$, i.e. ₹35 per unit.

**Verify MR = MC (dual check).**
$$	ext{MR} = \frac{dR}{dx} = 50 - 2x = 50 - 2(15) = 20, \qquad 	ext{MC} = \frac{dC}{dx} = 20.$$ …

Q.A firm faces the demand $p = 50 - x$ (price in ₹ when $x$ units are sold) and total cost $C(x) = 100 + 20x$ . Find the output that maximises profit, the maximum profit, and the price charged. Verify that MR = MC at this output.

Concept understanding — Optimisation of Profit and Cost

Q.A firm faces the demand p=50−xp = 50 - xp=50−x (price in ₹ when xxx units are sold) and total cost C(x)=100+20xC(x) = 100 + 20xC(x)=100+20x. Find the output that maximises profit, the maximum profit, and the price charged. Verify that MR = MC at this output.

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Q.A firm faces the demand $p = 50 - x$ (price in ₹ when $x$ units are sold) and total cost $C(x) = 100 + 20x$ . Find the output that maximises profit, the maximum profit, and the price charged. Verify that MR = MC at this output.

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