Q.The total cost function $y$ for $x$ units is given by $y = 3x\left(\dfrac{x + 7}{x + 5}\right) + 5$. Show that the Marginal Cost [MC] decreases continuously as the output $(x)$ increases.

Marginal cost is the rate of change of total cost with respect to output: $MC = \dfrac{dC}{dx}$, i.e. the cost of producing one additional unit.

GSEB Class-12 Statistics, Differentiation (business applications):

Marginal cost is the approximate change in total cost when the level of production is increased by one unit. If the total cost of producing $x$ units is $C = C(x)$, then marginal cost is the derivative of the total cost function:

$MC = \frac{dC}{dx}$

…

For $y=x^3+19$, the marginal value equals $27$ when $x=\pm 3$.

The marginal value is the first derivative of the function:

$\frac{dy}{dx}=\frac{d}{dx}\left(x^3+19\right)=3x^2.$

…

From $MR=AR\left(1-\tfrac1{\eta_d}\right)$, solving for $\eta_d$ yields $\eta_d=\dfrac{AR}{AR-MR}$.

The standard relationship between marginal revenue $(MR)$, average revenue $(AR)$ and price elasticity of demand $(\eta_d)$ is

$MR = AR\left(1 - \frac{1}{\eta_d}\right)$

Rearrange to isolate $\eta_d$:

…