Certainly! Let's analyze the cost function [tex]\( C(x) = 0.1x^2 - 7x + 140 \)[/tex] to find the number of golf balls [tex]\( x \)[/tex] that minimizes the production cost. Here's a step-by-step solution:
1. Identify the cost function:
The given cost function is:
[tex]\[
C(x) = 0.1x^2 - 7x + 140
\][/tex]
2. Find the critical points:
To find the critical points, we need to take the first derivative of the cost function [tex]\( C(x) \)[/tex] and set it to zero:
[tex]\[
C'(x) = \frac{d}{dx}(0.1x^2 - 7x + 140) = 0.2x - 7
\][/tex]
Set the first derivative equal to zero to solve for [tex]\( x \)[/tex]:
[tex]\[
0.2x - 7 = 0
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{7}{0.2} = 35
\][/tex]
So, the critical point is at [tex]\( x = 35 \)[/tex].
3. Classify the critical point:
To determine whether this critical point is a minimum or maximum, we take the second derivative of the cost function [tex]\( C(x) \)[/tex]:
[tex]\[
C''(x) = \frac{d^2}{dx^2}(0.1x^2 - 7x + 140) = 0.2
\][/tex]
The second derivative [tex]\( C''(x) = 0.2 \)[/tex] is positive, meaning the function [tex]\( C(x) \)[/tex] is concave upwards at [tex]\( x = 35 \)[/tex]. Hence, [tex]\( x = 35 \)[/tex] corresponds to a minimum point.
4. Calculate the minimum cost:
Substitute [tex]\( x = 35 \)[/tex] back into the original cost function to find the minimum cost:
[tex]\[
C(35) = 0.1(35)^2 - 7(35) + 140
\][/tex]
[tex]\[
C(35) = 0.1(1225) - 245 + 140
\][/tex]
[tex]\[
C(35) = 122.5 - 245 + 140
\][/tex]
[tex]\[
C(35) = 17.5
\][/tex]
Therefore, the number of golf balls [tex]\( x \)[/tex] that minimizes the daily production cost is [tex]\( x = 35 \)[/tex], and the minimum cost is [tex]\( C(35) = 17.5 \)[/tex].
In summary:
- The critical point is [tex]\( x = 35 \)[/tex].
- The second derivative at the critical point is positive ([tex]\( C''(x) = 0.2 \)[/tex]), confirming that it is a minimum.
- The minimum production cost is [tex]\( \$17.5 \)[/tex] when 35 golf balls are produced per hour.
