Explanation:
C(x) = 100 + 30x + (270/x)
A) C'(x) = 30 − (270/x²)
0 = 30 − (270/x²)
270/x² = 30
x² = 9
x = -3 or 3
B) x represents the production level per day, so it must be a positive number.
C) If you wish to use first derivative test, show that C'(x) changes signs around x = 3 from negative to positive, thus making x = 3 a minimum.
C'(2) = -37.5
C'(4) = 13.125
Therefore, x = 3 is a minimum.
If you wish to use second derivative test, show that C"(x) is positive at x = 3, thus making x = 3 a minimum.
C"(x) = 540/x³
C"(3) = 20
Therefore, x = 3 is a minimum.