To find the number of bicycles [tex]\( x \)[/tex] that Aki's Bicycle Designs should build to minimize the average cost per bicycle, we need to find the minimum point of the given cost function [tex]\( C(x) = 0.2x^2 - 1.4x + 4.643 \)[/tex].
Here's a step-by-step method to find this minimum point:
1. Identify the Function and its Derivative:
The cost function given is [tex]\( C(x) = 0.2x^2 - 1.4x + 4.643 \)[/tex].
To find the minimum point of this function, we first need to find the derivative of [tex]\( C(x) \)[/tex], often denoted as [tex]\( C'(x) \)[/tex].
2. Take the First Derivative:
Compute the first derivative of [tex]\( C(x) \)[/tex]:
[tex]\[
C'(x) = \frac{d}{dx}(0.2x^2 - 1.4x + 4.643) = 0.4x - 1.4
\][/tex]
3. Set the Derivative to Zero:
To find the critical points, we set [tex]\( C'(x) \)[/tex] to zero and solve for [tex]\( x \)[/tex]:
[tex]\[
0.4x - 1.4 = 0
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
Solve the equation [tex]\( 0.4x - 1.4 = 0 \)[/tex]:
[tex]\[
0.4x = 1.4
\][/tex]
[tex]\[
x = \frac{1.4}{0.4}
\][/tex]
[tex]\[
x = 3.5
\][/tex]
5. Interpret the Result:
The critical point [tex]\( x = 3.5 \)[/tex] shows the number of hundreds of bicycles that minimize the average cost per bicycle.
Since [tex]\( x = 3.5 \)[/tex] represents 3.5 hundred bicycles:
The shop should build [tex]\( 3.5 \times 100 = 350 \)[/tex] bicycles to minimize the average cost per bicycle.
Thus, the shop should build [tex]\( \boxed{350} \)[/tex] bicycles.
