Marginal cost curve intersects average cost curve at _____________

The correct option is: At minimum point from of average cost curve.

Explanation:
In microeconomics, the relationship between the Average Cost (AC) curve and the Marginal Cost (MC) curve is a fundamental concept determined by mathematical logic and cost dynamics. We can understand why the Marginal Cost curve intersects the Average Cost curve at its lowest (minimum) point through both intuitive economic reasoning and mathematical derivation.

1. Intuitive Economic Reasoning:
The relationship between Average Cost and Marginal Cost behaves similarly to cumulative averages in everyday life (such as a student's grade point average):

• When MC < AC: If the cost of producing one additional unit (Marginal Cost) is less than the current average cost, it pulls the average cost down. Thus, the Average Cost curve is downward-sloping.
• When MC > AC: If the cost of producing one additional unit (Marginal Cost) is greater than the current average cost, it pulls the average cost up. Thus, the Average Cost curve begins to rise and is upward-sloping.
• At the Minimum Point of AC: At the transition point where the average cost stops falling and is about to start rising (which is the minimum point of the AC curve), the marginal cost must be exactly equal to the average cost. Therefore, the MC curve intersects the AC curve at this exact minimum point.

2. Mathematical Derivation:
Let Total Cost be represented by $TC$ and the quantity of output by $q$. The Average Cost ($AC$) is defined as:

$$AC=\frac{TC}{q}$$

To find the quantity $q$ that minimizes the Average Cost, we take the derivative of $AC$ with respect to $q$ and set it equal to zero:

$$\frac{d\left(AC\right)}{dq}=0$$

Applying the quotient rule of differentiation to $\frac{TC}{q}$, we get:

$$\frac{q·\frac{d\left(TC\right)}{dq}-TC}{q^2}=0$$

By definition, Marginal Cost ($MC$) is the derivative of Total Cost with respect to quantity, i.e., $MC=\frac{d\left(TC\right)}{dq}$. Substituting $MC$ into the equation:

$$\frac{q·MC-TC}{q^2}=0$$

Multiplying both sides by $q^2$ (assuming $q>0$):

$$q·MC-TC=0$$

Rearranging the terms:

$$q·MC=TC$$

Dividing by $q$:

$$MC=\frac{TC}{q}$$

Since $\frac{TC}{q}=AC$, we have:

$$MC=AC$$

This mathematical result proves that at the minimum point of the Average Cost curve, Marginal Cost is exactly equal to Average Cost, meaning the curves intersect at this point.