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Solve the following.
The cost of manufacturing x units of a commodity is 27 + 12x + 3x². Find the output for which…
The cost of manufacturing $x$ units of a commodity is $27 + 12x + 3x^2$. Find the output for which Average Cost is decreasing.
Answer
Answer
$AC = \frac{27 + 12x + 3x^2}{x} = \frac{27}{x} + 12 + 3x$.
$\frac{d(AC)}{dx} = -\frac{27}{x^2} + 3$.
AC is decreasing when $-\frac{27}{x^2} + 3 < 0 \Rightarrow x^2 < 9 \Rightarrow 0 < x < 3$ (output is positive).
Final answer: $0 < x < 3$