Question

a company that produces cell phones has a cost function of $c = 7x^{2}-144x + 17689$, where $c$ is the cost in dollars and $x$ is the number of cell phones produced (in thousands). how many units of cell phone (in thousands) minimizes the cost function? $x=$ thousand phones produced will minimize $c$

Explanation:

Step1: Find the derivative of the cost function

The cost function is \(C = 7x^{2}-144x + 17689\).
Using the power rule \((x^{n})^\prime=nx^{n - 1}\), the derivative \(C^\prime(x)\) is:
\(C^\prime(x)=\frac{d}{dx}(7x^{2}-144x + 17689)=7\times2x-144+0 = 14x-144\)

Step2: Set the derivative equal to zero and solve for \(x\)

To find the critical points, set \(C^\prime(x)=0\):
\(14x - 144=0\)
Add \(144\) to both sides: \(14x=144\)
Divide both sides by \(14\): \(x=\frac{144}{14}=\frac{72}{7}\approx10.29\)

Step3: Check the second - derivative

The second - derivative \(C^{\prime\prime}(x)=\frac{d}{dx}(14x - 144)=14>0\)
Since \(C^{\prime\prime}(x)>0\), the function \(C(x)\) is concave up at \(x = \frac{72}{7}\), and this critical point is a minimum.

Answer:

\(\frac{72}{7}\)