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Answer the following.
For the revenue function R(x) = (bx)/(a+x), show that the marginal revenue function is increasing…
For the revenue function $R(x) = \frac{bx}{a+x}$, show that the marginal revenue function is increasing for all $b < 0$ and $a > 0$.
Given: $R(x) = \frac{bx}{a+x}$
To show: marginal revenue function is increasing for all $b < 0$ and $a > 0$
Answer
Answer
Differentiate $R(x)$ to get $MR$, then show $\frac{d(MR)}{dx} > 0$.
- $R(x) = \frac{bx}{a+x}$. By the quotient rule, $MR = \frac{dR}{dx} = \frac{b(a+x) - bx}{(a+x)^2} = \frac{ab}{(a+x)^2}$.
- Differentiating $MR$ again: $\frac{d(MR)}{dx} = ab\cdot(-2)(a+x)^{-3} = -\frac{2ab}{(a+x)^3}$.
- Given $b < 0$ and $a > 0$, $ab < 0$, so $-2ab > 0$.
- For positive $x$ and $a > 0$, $(a+x)^3 > 0$.
- Hence $\frac{d(MR)}{dx} > 0$, so the marginal revenue function is increasing for all $b < 0$ and $a > 0$.