Which of the following best describes the relationship between the average total cost curve and the marginal cost curve in the shown?

Question

BlackTom AI · Accepted Answer

Differentiate average total cost (ATC = TC/Q) with respect to output Q: d(ATC)/dQ = (MC − ATC)/Q. This key relation links marginal cost (MC) and the slope of the ATC curve.

1) Relevant concepts
- ATC = TC/Q, MC = d(TC)/dQ.
- The sign of d(ATC)/dQ determines whether ATC is rising (positive), falling (negative) or flat (zero).
- The formula d(ATC)/dQ = (MC − ATC)/Q shows that the difference (MC − ATC) determines the direction of ATC.

2) Step-by-step reasoning
- Starting expression: ATC(Q) = TC(Q)/Q.
- Differentiate using quotient rule or the known identity: d(ATC)/dQ = (Q·MC − TC)/Q^2 = (MC − TC/Q)/Q = (MC − ATC)/Q.
- Because Q > 0, the sign of d(ATC)/dQ is the same as the sign of (MC − ATC).
- If ATC is rising: d(ATC)/dQ > 0, so (MC − ATC) > 0 ⇒ MC > ATC. Thus the marginal cost curve lies above the ATC curve when ATC is increasing.
- If ATC is falling: MC < ATC (marginal lies below ATC). If ATC is flat (minimum), MC = ATC (they intersect).

3) No numeric calculation is required beyond the derivative; intermediate algebra shown above.

4) Why the correct answer is correct
The correct option states: "If the average total cost curve is rising, the marginal cost curve is above the average total cost curve." This follows directly because a rising ATC means positive derivative, which requires MC − ATC > 0, i.e. MC > ATC.

5) Why other options are incorrect
- Option 2 (MC below ATC when ATC is rising): opposite of the true relation; wrong sign.
- Option 3 (If ATC above MC, MC is rising): the spatial relation (MC below ATC) does not imply the slope (rising) of MC; position vs. slope are different properties.
- Option 4 (If ATC below MC, MC is falling): also confuses position with slope; MC being above ATC implies ATC rising, not that MC is falling.
- Option 5 (If ATC and MC intersect, the marginal cost curve is at a minimum): intersection means MC = ATC and typically marks the minimum of ATC, not necessarily the minimum of MC; MC’s own minimum generally occurs at a different quantity.

Conclusion: Therefore the correct statement is that when ATC is rising, MC is above ATC, which matches the provided answer.

Which of the following best describes the relationship between the average total cost curve and the marginal cost curve in the shown?单项选择题

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