Given that C(x)=2x^3-21x^2+36x+1000 is a cost function, determine the intervals for which the cost is increasing. Determine any intervals for which the marginal cost is increasing.
Marginal cost is when you derive the cost function, correct? So do you do the second derivative for the marginal cost intervals?
(So far, I just did the first derivative and got negative infinity to 1 and 6 to infinity to be my answers, but are those right or wrong, and would they count for marginal cost or regular cost?)
To determine the intervals for which the cost is increasing, we need to find the critical points (where the derivative of the cost function is equal to zero or undefined) and then determine the sign of the derivative in the intervals between the critical points.
Now let's find the derivative of the cost function C(x):
C'(x) = d/dx(2x^3 - 21x^2 + 36x + 1000)
C'(x) = 6x^2 - 42x + 36
To find the critical points, we set the derivative equal to zero and solve for x:
6x^2 - 42x + 36 = 0
Dividing both sides by 6:
x^2 - 7x + 6 = 0
Factoring the quadratic equation:
(x - 1)(x - 6) = 0
So the critical points are x = 1 and x = 6.
Now, let's consider the intervals between the critical points and determine the sign of the derivative in each interval.
For x < 1, we can pick a test point, such as x = 0, and substitute it into the derivative:
C'(0) = 6(0)^2 - 42(0) + 36 = 36
Since the derivative is positive, the cost function is increasing in this interval.
For 1 < x < 6, we can pick a test point, such as x = 3, and substitute it into the derivative:
C'(3) = 6(3)^2 - 42(3) + 36 = -18
Since the derivative is negative, the cost function is decreasing in this interval.
For x > 6, we can pick a test point, such as x = 7, and substitute it into the derivative:
C'(7) = 6(7)^2 - 42(7) + 36 = 144
Since the derivative is positive, the cost function is increasing in this interval.
Therefore, the intervals where the cost is increasing are x < 1 and x > 6.
Now, let's determine intervals for which the marginal cost is increasing. Yes, you are correct! Marginal cost is obtained by taking the derivative of the cost function. To determine intervals where the marginal cost is increasing, we need to find the critical points of the derivative function (the second derivative of the cost function) and then determine the sign of the second derivative in those intervals.
Let's find the second derivative of the cost function C'(x):
C''(x) = d^2/dx^2(6x^2 - 42x + 36)
C''(x) = 12x - 42
Now, to find the critical points of the second derivative, we set it equal to zero and solve for x:
12x - 42 = 0
Dividing both sides by 12 gives:
x = 3.5
Since we only have one critical point, we only need to check one interval.
For x < 3.5, we can pick a test point, such as x = 2, and substitute it into the second derivative:
C''(2) = 12(2) - 42 = -18
Since the second derivative is negative, the marginal cost is decreasing in this interval.
For x > 3.5, we can pick a test point, such as x = 4, and substitute it into the second derivative:
C''(4) = 12(4) - 42 = 6
Since the second derivative is positive, the marginal cost is increasing in this interval.
Therefore, the interval for which the marginal cost is increasing is x > 3.5.
In conclusion, the intervals for which the cost is increasing are x < 1 and x > 6, and the interval for which the marginal cost is increasing is x > 3.5.