a vehicle factory manufactures cars. the unit cost c (the cost in dollars to make each car) depends on the…

a vehicle factory manufactures cars. the unit cost c (the cost in dollars to make each car) depends on the number of cars made. if x cars are made, then the unit cost is given by the function c(x)=0.4x² - 96x + 11,904. what is the minimum unit cost? do not round your answer. unit cost: $

a vehicle factory manufactures cars. the unit cost c (the cost in dollars to make each car) depends on the number of cars made. if x cars are made, then the unit cost is given by the function c(x)=0.4x² - 96x + 11,904. what is the minimum unit cost? do not round your answer. unit cost: $

Answer

Explanation:

Step1: Recall derivative - minimum concept

To find the minimum of a function (y = f(x)), we first find its derivative (y'=f'(x)) and then set (y' = 0) to find the critical - points. For the cost function (C(x)=0.4x^{2}-96x + 11904), its derivative (C'(x)) is calculated using the power rule ((x^{n})'=nx^{n - 1}).

Step2: Calculate the derivative of (C(x))

[ \begin{align*} C'(x)&=\frac{d}{dx}(0.4x^{2}-96x + 11904)\ &=0.4\times2x-96+0\ &=0.8x - 96 \end{align*} ]

Step3: Find the critical - point

Set (C'(x)=0), then (0.8x - 96 = 0). Solve for (x): [ \begin{align*} 0.8x&=96\ x&=\frac{96}{0.8}=120 \end{align*} ]

Step4: Determine if it's a minimum

We can find the second - derivative (C''(x)) to determine if the critical - point is a minimum. (C''(x)=\frac{d}{dx}(0.8x - 96)=0.8>0). Since (C''(x)>0), (x = 120) is a point of minimum for the cost function (C(x)).

Step5: Calculate the minimum unit cost

Substitute (x = 120) into the cost function (C(x)): [ \begin{align*} C(120)&=0.4\times(120)^{2}-96\times120 + 11904\ &=0.4\times14400-11520 + 11904\ &=5760-11520 + 11904\ &=6144 \end{align*} ]

Answer:

6144