Answer:
The minimum average cost is $2.80 when 163 players are built
Step-by-step explanation:
Average Cost Function
We'll assume the given function as the total cost to produce x players, and NOT the average cost since that is a different definition, as shown below.
The cost C to produce x DVD/Blu- ray players is given by the equation
[tex]C=0.03x^2-7x+800[/tex]
The Average Cost function is defined as
[tex]\displaystyle\bar C=\frac{C}{x}[/tex]
[tex]\displaystyle\bar C=\frac{0.03x^2-7x+800}{x}[/tex]
[tex]\displaystyle\bar C=0.03x-7+\frac{800}{x}[/tex]
To find the extreme value of the average cost, we must take the first derivative of the function
[tex]\displaystyle\bar C'=0.03-\frac{800}{x^2}[/tex]
Equating to 0
[tex]\displaystyle 0.03-\frac{800}{x^2}=0[/tex]
Solving for x
[tex]\displaystyle x=\sqrt{\frac{800}{0.03}}[/tex]
[tex]x=163[/tex]
The second derivative is
[tex]\displaystyle\bar C''=\frac{1600}{x^3}[/tex]
For x=163 the second derivative is positive, thus x=163 is a minimum value. Let's compute the minimum average cost
[tex]\displaystyle\bar C(163)=0.03\cdot 163-7+\frac{800}{163}[/tex]
[tex]\displaystyle\bar C(163)=2.80[/tex]
The minimum average cost is $2.80
