The cost function at American Gadget is C(x) | vedclass

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(B) The cost function is given by $C(x) = x^3 - 6x^2 + 15x$.The average cost function $AC(x)$ is defined as $\frac{C(x)}{x}$.$AC(x) = \frac{x^3 - 6x^2

Vedclass · Accepted Answer

$3$

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(B) The cost function is given by $C(x) = x^3 - 6x^2 + 15x$.The average cost function $AC(x)$ is defined as $\frac{C(x)}{x}$.$AC(x) = \frac{x^3 - 6x^2 + 15x}{x} = x^2 - 6x + 15$ for $x > 0$.To find the minimum average cost,we find the derivative of $AC(x)$ with respect to $x$ and set it to zero:$\frac{d}{dx}(AC(x)) = \frac{d}{dx}(x^2 - 6x + 15) = 2x - 6$.Setting the derivative to zero:$2x - 6 = 0 \implies 2x = 6 \implies x = 3$.To verify this is a minimum,we check the second derivative:$\frac{d^2}{dx^2}(AC(x)) = 2$.Since the second derivative is positive $(2 > 0)$,the function $AC(x)$ has a local minimum at $x = 3$.

The cost function at American Gadget is $C(x) = x^3 - 6x^2 + 15x$ (where $x$ is in thousands of units and $x > 0$). The production level at which the average cost is minimum is:

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