suppose that ( c(x)=3x^{3}-18x^{2}+14,000x ) is the cost of manufacturing ( x ) items. find a production…

suppose that ( c(x)=3x^{3}-18x^{2}+14,000x ) is the cost of manufacturing ( x ) items. find a production level that will minimize the average cost of making ( x ) items. the production level that minimizes the average cost of making ( x ) items is ( x=) (simplify your answer.)

suppose that ( c(x)=3x^{3}-18x^{2}+14,000x ) is the cost of manufacturing ( x ) items. find a production level that will minimize the average cost of making ( x ) items. the production level that minimizes the average cost of making ( x ) items is ( x=) (simplify your answer.)

Answer

Explanation:

Step1: Find the average cost function

The average cost function ( \overline{c}(x)=\frac{c(x)}{x} ). Given ( c(x) = 3x^{3}-18x^{2}+14000x ), then ( \overline{c}(x)=\frac{3x^{3}-18x^{2}+14000x}{x}=3x^{2}-18x + 14000).

Step2: Differentiate the average cost function

Differentiate ( \overline{c}(x) ) with respect to ( x ). Using the power rule ( (x^{n})^\prime=nx^{n - 1} ), we have ( \overline{c}^\prime(x)=(3x^{2}-18x + 14000)^\prime=6x-18).

Step3: Find the critical points

Set ( \overline{c}^\prime(x)=0 ), so ( 6x-18 = 0 ). Solving for ( x ): [ \begin{align*} 6x&=18\ x&=3 \end{align*} ]

Step4: Check the second - derivative

Differentiate ( \overline{c}^\prime(x) ) to get the second - derivative. ( \overline{c}^{\prime\prime}(x)=(6x - 18)^\prime=6>0 ). Since ( \overline{c}^{\prime\prime}(3)=6>0 ), the function ( \overline{c}(x) ) has a minimum at ( x = 3 ).

Answer:

(3)