Q.The price 'p' per unit is given by the relation $x = \dfrac{1}{3}p^2 - 2p + 3$ where 'x' is the number of units sold then, i. Find the revenue function R.
ii. Find the price interval for which the revenue is increasing and decreasing.

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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