consider a manufacture whose total cost of producing x items is given by c(x) = 10000 + 5x+1/9x^2
a) what is the average cost function of A(x) = c(x)/x?
b)how many items should the manufacturer produce in order to minimize average cost?
c)what is the smallest average cost?
d) find c(x)
e) when does c(x) have a critical point? what is the average cost when c(x) has a critical point?
a(x) = c(x)/x = 10000/x + 5 + 1/9 x
da/dx = -1000/x^2 + 1/9
da/dx=0 when x = 30√10 = 94.868
a(94) = 121.8
a(95) = 120.8
so, it looks like the lowest average cost is at x=95
c(x) is a parabola whose vertex is at a negative x-value. So, there is no minimum cost.
To answer these questions, we will follow the given steps:
a) The average cost function (A(x)) can be found by dividing the total cost (c(x)) by the number of items produced (x). So, the average cost function is A(x) = c(x)/x. We substitute the given expression for c(x): A(x) = (10000 + 5x + (1/9)x^2)/x.
b) To find the number of items that minimize the average cost, we need to find the value of x that minimizes the average cost function A(x). We can do this by differentiating A(x) with respect to x and setting it equal to zero. So, we differentiate A(x) and set it equal to zero: dA/dx = (1/9)x^2 - (5/9)x - 10000/x^2 = 0. Solving this equation will give us the value of x that minimizes the average cost.
c) Once we find the value of x that minimizes the average cost in part b, we can substitute this value back into the average cost function A(x) to find the corresponding average cost.
d) To find c(x), we use the expression for c(x) given in the question: c(x) = 10000 + 5x + (1/9)x^2.
e) To find the critical points of c(x), we need to differentiate c(x) with respect to x and set it equal to zero. So, we differentiate c(x) and set it equal to zero: dc/dx = 5 + (2/9)x = 0. Solving this equation will give us the critical point(s) of c(x). Once we find the critical point(s), we can substitute these values into the average cost function A(x) to find the corresponding average cost.
To summarize:
a) A(x) = (10000 + 5x + (1/9)x^2)/x
b) Find the value of x that makes dA/dx = 0 and substitute it into A(x) to find the minimum average cost.
c) Substitute the value of x from part b into A(x) to find the minimum average cost.
d) c(x) = 10000 + 5x + (1/9)x^2
e) Solve dc/dx = 0 to find the critical point(s) of c(x) and substitute them into A(x) to find the corresponding average cost.