Q.If price 'p' per unit of an article is p = 75 – 2x and the cost function is $C(x) = 350 + 12x + \dfrac{x^2}{4}$. Find the number of units and the price at which the total profit is maximum. What is the maximum profit?

Total Cost Function: From Intuition to Definition

Imagine you run a small stall selling lemonade. You need to buy lemons, sugar, cups, and ice — those are your variable costs, because they change with how many glasses you sell. But you also paid for the stall license, the table, and the pitcher — those are fixed costs, because they stay the same whether you sell one glass or a hundred.

The total cost is simply everything you spent: fixed costs plus variable costs. If you sell $Q$ glasses, your total cost is:

where $FC$ is fixed cost (constant) and $VC(Q)$ is variable cost (depends on quantity $Q$).

That is the intuition. Now the precise statement.

The Formal Definition

In economics, the Total Cost Function $TC(Q)$ gives the minimum cost of producing $Q$ units of output, given the prices of inputs (like labour and raw materials) and the available technology. It is always a function of output $Q$, and it is defined for all non-negative $Q$.

Key properties you will encounter:

• $TC(0) = TFC$ — even with zero output, fixed costs are incurred.
• $TC(Q)$ is increasing in $Q$ — more output always costs more (or at least not less).
• The shape of $TC(Q)$ depends on the production technology: it may be linear, concave, or convex.

Why "Function"?

The word "function" is deliberate. For every possible output level $Q$, there is exactly one total cost. This lets us do calculus — find marginal cost, average cost, and so on. In Indian board exams (CBSE Class 12 Economics), you will be asked to:

• Derive $TC$ from a table of $TFC$ and $TVC$.
• Draw the $TC$ curve alongside $TFC$ and $TVC$.
• Compute $TC$ given $MC$ (marginal cost) and initial fixed cost.

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