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a manufacturing firm has a total cost function given by c x 2000 500x 0 5xsquared where x is the number of units produced

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A manufacturing firm has a total cost function given by C(x)= 2000+500x +0.5xsquared where x is the number of units produced, how many units must be produced to minimize the average cost?

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Answer to a math question A manufacturing firm has a total cost function given by C(x)= 2000+500x +0.5xsquared where x is the number of units produced, how many units must be produced to minimize the average cost?

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El costo promedio \( \bar{C}(x) \) es:

\bar{C}(x) = \frac{2000}{x} + 500 + 0.5x

Derivando e igualando a cero:

\frac{d\bar{C}(x)}{dx} = -\frac{2000}{x^2} + 0.5 \quad \Rightarrow \quad -\frac{2000}{x^2} + 0.5 = 0

Resolviendo para \( x \):

2000 = 0.5x^2 \quad \Rightarrow \quad x^2 = 4000 \quad \Rightarrow \quad x = 20\sqrt{10}

Por lo tanto, el número de unidades que minimiza el costo promedio es:

x = 20\sqrt{10}

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