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Answer the following.
The total cost function for x units is given by C(x) = √(6x + 5) + 2500. Show that the marginal…
The total cost function for $x$ units is given by $C(x) = \sqrt{6x + 5} + 2500$. Show that the marginal cost decreases as the output $x$ increases.
Given: $C(x) = \sqrt{6x + 5} + 2500$
To show: $\text{Marginal cost decreases as the output } x \text{ increases}$
Answer
Answer
Marginal cost $MC = \frac{dC}{dx} = \frac{3}{\sqrt{6x+5}}$. Its rate of change is negative for all $x > 0$, so $MC$ decreases as $x$ increases.
- Step 1: Given $C(x) = \sqrt{6x+5} + 2500$.
- Step 2: Marginal cost $MC = \frac{dC}{dx} = \frac{6}{2\sqrt{6x+5}} = \frac{3}{\sqrt{6x+5}}$ (chain rule).
- Step 3: Differentiating $MC = 3(6x+5)^{-1/2}$: $\frac{d(MC)}{dx} = 3\cdot\left(-\frac{1}{2}\right)(6x+5)^{-3/2}\cdot 6 = -\frac{9}{(6x+5)^{3/2}}$.
- Step 4: For $x \geq 0$, $(6x+5)^{3/2} > 0$, so $\frac{d(MC)}{dx} < 0$.
- Step 5: Hence the marginal cost decreases as the output $x$ increases.