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Prove the following.
The total variable cost of manufacturing units in a firm is ₹ . Show that average variable cost…
The total variable cost of manufacturing $x$ units in a firm is ₹ $(3x + \frac{x^5}{25})$. Show that average variable cost increases with output $x$.
Given: The total variable cost of manufacturing $x$ units in a firm is ₹ $(3x + \frac{x^5}{25})$
To show: Average variable cost increases with output $x$
Answer
Answer
AIAverage variable cost $AVC = \frac{3x + \frac{x^5}{25}}{x} = 3 + \frac{x^4}{25}$.
$\frac{d}{dx}(AVC) = \frac{4x^3}{25}$.
For $x > 0$, $\frac{4x^3}{25} > 0$.
Hence the average variable cost increases with output $x$.
- $AVC = \frac{\text{total variable cost}}{x} = 3 + \frac{x^4}{25}$.
- $\frac{d(AVC)}{dx} = \frac{4x^3}{25}$.
- For output $x > 0$, $\frac{4x^3}{25} > 0$, so $AVC$ is increasing.
- Hence the average variable cost increases with output $x$.
From ISC 2018 Specimen Mathematics Paper 1, question 19(c).