Suppose that $$ c(x)=3 x^{3}-18 x^{2}+24,000 x $$ is the cost of manufacturing $$ x $$ items. Find a production level that will minimize the average cost of making $$ x $$ items. At this level, the average cost of making $$ x $$ items is $$ \$ $$ (Simplify your answer.)

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The average cost of making $$ x $$ items at the production level that minimizes the average cost is $$ \$23973 $$. To find the production level that minimizes the average cost of making $$ x $$ items, we first need to determine the average cost function. The average cost $$ A(x) $$ is given by the total cost $$ c(x) $$ divided by the number of items $$ x $$: $$ A(x) = \frac{c(x)}{x} = \frac{3x^3 - 18x^2 + 24000x}{x} $$ Simplifying this, we get: $$ A(x) = 3x^2 - 18x + 24000 $$ Next, we need to find the critical points of $$ A(x) $$ by taking the derivative and setting it to zero: $$ A'(x) = \frac{d}{dx}(3x^2 - 18x + 24000) = 6x - 18 $$ Setting the derivative equal to zero to find critical points: $$ 6x - 18 = 0 $$ Solving for $$ x $$: $$ 6x = 18 \implies x = 3 $$ To confirm that this critical point is a minimum, we can check the second derivative: $$ A''(x) = \frac{d^2}{dx^2}(3x^2 - 18x + 24000) = 6 $$ Since $$ A''(x) = 6 > 0 $$, the function $$ A(x) $$ is concave up at $$ x = 3 $$, indicating that this point is indeed a minimum. Now, we can find the average cost at this production level: $$ A(3) = 3(3^2) - 18(3) + 24000 $$ Calculating each term: $$ A(3) = 3(9) - 54 + 24000 = 27 - 54 + 24000 = -27 + 24000 = 23973 $$ Thus, the average cost of making $$ x $$ items at the production level that minimizes the average cost is: $$ \boxed{23973} $$

Suppose that \( c(x)=3 x^{3}-18 x^{2}+24,000 x \) is the cost of manufacturing \( x \) items. Find a production level that will minimize the average cost of making \( x \) items. At this level, the average cost of making \( x \) items is \( \$ \) (Simplify your answer.)

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Suppose that \( c(x)=3 x^{3}-18 x^{2}+24,000 x \) is the cost of manufacturing \( x \) items. Find a production level that will minimize the average cost of making \( x \) items. At this level, the average cost of making \( x \) items is \( \$ \) (Simplify your answer.)

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