Q.For what values of x is the rate of increase of total cost function $C(x) = x^3 - 5x^2 + 5x + 8$ is twice the rate of increase of x?

Marginal Cost Minimization: The Intuition First

Imagine you run a small bakery. Every morning you bake a batch of 100 loaves of bread. Your total cost — rent, flour, electricity, your own time — comes to ₹2000. That's an average cost of ₹20 per loaf.

Now suppose a customer walks in and asks for one extra loaf. You have to turn the oven back on, use a little more flour, and stay an extra 15 minutes. That single extra loaf costs you, say, ₹25 in additional flour and electricity. That ₹25 is the marginal cost — the cost of producing one more unit.

Here's the key question: At what level of production does that marginal cost become as low as possible?

That's what marginal cost minimization is about. It's not about making the total cost small — it's about finding the sweet spot where the next unit costs you the least to produce.

The Precise Statement

Marginal Cost (MC) is the change in total cost when output increases by one unit:

Marginal Cost Minimization means finding the output level $Q$ at which this marginal cost is at its lowest value.

Why does this happen? In the short run, production typically follows a pattern:

1. Increasing returns (falling MC): As you hire more workers or use more inputs, specialization kicks in. Each extra unit costs less than the previous one.
2. Diminishing returns (rising MC): Eventually, adding more inputs becomes crowded and inefficient. Each extra unit starts costing more.

The minimum marginal cost occurs right at the boundary between these two phases — where the law of diminishing returns begins to set in.

The Mathematical Picture

If the total cost function is $TC(Q)$, then:

To minimize $MC$, we set its derivative to zero:

This is the second derivative of the total cost function being zero — the inflection point.

A Concrete Example

Suppose a factory's total cost (in ₹) for producing $Q$ units is:

Then marginal cost is:

To minimize MC, differentiate and set to zero:

At $Q = 4$, the marginal cost is:

So the minimum marginal cost is ₹12 per unit, achieved at an output of 4 units. …