The cost to produce x units of a certain product is given by $C ( x ) = 10,000 + 8 x + \frac { 1 } { 16 } x ^ { 2 }$ . Find the value of x that gives the minimum average cost.
A)400
B)25
C)160,000
D)4000
E)500
F)64
G)40,000
H)Does not exist

Question

Quizplus · Accepted Answer

The average cost, $AC$, is given by $AC(x) = \frac{C(x)}{x} = \frac{10,000 + 8x + \frac{1}{16}x^2}{x}$. To find the minimum average cost, we differentiate $AC(x)$ with respect to $x$ and set the derivative equal to zero. The derivative of $AC(x)$ is $AC'(x) = 8 + \frac{1}{8}x - \frac{10,000}{x^2}$. Setting $AC'(x) = 0$ and solving for $x$ gives $x = 400$, which is the value of $x$ that minimizes the average cost.

The Cost to Produce X Units of a Certain Product C(x)=10,000+8x+116x2C ( x ) = 10,000 + 8 x + \frac { 1 } { 16 } x ^ { 2 }C(x)=10,000+8x+161​x2

The Cost to Produce X Units of a Certain Product $C ( x ) = 10,000 + 8 x + \frac { 1 } { 16 } x ^ { 2 }$