Respuesta :
At x = 1000 we get the c(1000) = 2000,000. Second statement 'one thousand units have the maximum cost of $200,000' is true.
It is given that the function [tex]\rm c(x)=400x-0.2x^2[/tex] represents the total costs for a company to produce a product, where c is the total cost in dollars and x is the number of units sold.
It is required to find which statement is true, mentioned in the question.
What are the maxima and minima of a function?
Maxima and minima of a function are the extrema within the range, in other words, the maximum value of a function at a certain point is called maxima and the minimum value of a function at a certain point is called minima.
We have a function:
[tex]\rm c(x)=400x-0.2x^2[/tex]
To find the maxima and minima of any function we differentiate the function with respect to [tex]\rm x[/tex] and equate to zero.
[tex]\rm \frac{d \ c(x)}{dx} =\frac{d}{dx} (400x-0.2x^2)[/tex]
[tex]\rm \frac{d \ c(x)}{dx} =\frac{d}{dx} (400x)-\frac{d}{dx} (0.2x^2)\\\rm \frac{d \ c(x)}{dx} =400-(0.2\times2x)\\\rm \frac{d \ c(x)}{dx} =400-0.4x[/tex]
For maxima and minima [tex]\rm \frac{d \ c(x)}{dx} =0[/tex]
[tex]\rm 400-0.4x=0\\\rm x=\frac{400}{0.4} \\\rm x=1000[/tex]
If we find the seconnd differentiate, we get
c''(x) = -0.4 < 0
It means at x = 1000 we will get the maximum value of a function.
[tex]\\\\\rm c(1000) = 400(1000) - 0.2(1000)^2\\\rm c(1000)= 200,000[/tex]
Thus, at x = 1000 we get the c(1000) = 2000,000. Second statement is true.
Know more about the maxima and minima here:
https://brainly.com/question/6422517
Answer:
B
Step-by-step explanation:
To determine the maxima and minima of the polynomial, differentiate the given based on x and equate to 0. C(x) = 400x - 0.2x² dC(x) / dt = 400 - 0.4 x = 0 The value of x is 1000. This is the value of the maxima. As the value of C(x) continously becomes lesser as the value of x is set higher, the minima is not identified. Substitute x to the original equation, C(x) = (400)(1000) - 0.2(1000²) = $ 200,000Thus, the answer is letter B.
