Minimizing Average Cost Suppose the total cost function for manufacturing a certain product C(x) is given by the function below, where C(x) is measured in dollars and x represents the number of units produced. Find the level of production (in units) that will minimize the average cost. (Round your answer to the nearest whole number.)
C(x) = 0.2(0.01x2 + 141)
___ units
g(x)= x sqrt (9-x^2) on [0,3]
Max:
To find the level of production that will minimize the average cost, we need to differentiate the cost function with respect to x and set it equal to zero. This is because the minimum occurs at the critical points where the derivative is zero.
First, let's differentiate the cost function, C(x), with respect to x:
C'(x) = 0.2 * d/dx (0.01x^2 + 141)
To differentiate the function 0.01x^2 + 141, we can use the power rule. According to the power rule, the derivative of x^n, where n is a constant, is nx^(n-1). Applying the power rule, we get:
C'(x) = 0.2 * (0.02x) = 0.004x
Now, let's set C'(x) equal to zero and solve for x:
0.004x = 0
Dividing both sides by 0.004, we get:
x = 0
Therefore, the critical point is x = 0.
However, we need to find a positive value for x since production cannot be zero. To determine if x = 0 is a minimum or maximum, we need to examine the second derivative.
To find the second derivative, we differentiate C'(x):
C''(x) = d/dx (C'(x)) = d/dx (0.004x) = 0.004
Since the second derivative is a constant (0.004), it tells us that the graph of C(x) is a parabola that opens upwards.
Since the second derivative is positive, it indicates that x = 0 is a minimum point for the cost function.
Therefore, the level of production that will minimize the average cost is x = 0 units.
as you know, average = total/size So avg cost A(x) is
A(x) = C(x)/x = 0.2 (0.01x + 141/x)
dA/dx = 0 at x = 10√141