Suppose the demand per month for a commodity is $24$ if the price is ₹ $16$ and $12$ if the price is ₹ $22$. Assuming the demand curve is linear, determine the demand function, the total revenue function and the marginal revenue function.
Let $p =$ price of a commodity
$x =$ demand per month for a commodity
Given: The demand curve is linear.
$\therefore \;$ Let the demand function be: $\;$ $p = a + bx$ $\;\;\; \cdots \; (1)$ $\;\;$ where $a$ and $b$ are constants
Given: $\;$ when $\;$ $x = 24$, $p = 16$; $\;$ when $\;$ $x = 12$, $p = 22$
Substituting the values of $p$ and $x$ in equation $(1)$ we have,
$16 = a + 24 b$ $\;\;\; \cdots \; (2a)$
$22 = a + 12 b$ $\;\;\; \cdots \; (2b)$
Solving equations $(2a)$ and $(2b)$ simultaneously we have,
$b = \dfrac{-1}{2}$ $\;$ and $\;$ $a = 28$
$\therefore \;$ The demand function is $\;$ $p = 28 - \dfrac{1}{2}x$
Total revenue function $R = p \cdot x$
i.e. $\;$ $R = \left(28 - \dfrac{1}{2} x\right) \times x$
i.e. $\;$ $R = 28 x - \dfrac{1}{2} x^2$
Marginal revenue function $= MR = \dfrac{dR}{dx}$
i.e. $\;$ $MR = \dfrac{d}{dx} \left(28 x - \dfrac{1}{2} x^2\right)$
i.e. $\;$ $MR = 28 - x$