Math Help Please :) ?

Question

The cost $C, for the production of x units of a certain product is given by C=(x+10)^3 Find the value of x for which the average cost per unit is a minimum and find this minimum average cost

Answer 1

Average cost per unit = f(x) = C/x
= x^2 + 30x + 300 + 1000/x
f'(x) = 2x + 30 - 1000/x^2
f"(x) = 2 + 2000/x^3

f'(x) = 0 <=> 2x^3 + 30x^2 - 1000 = 0
<=> 2(x+10)(x^2 + 5x - 50) = 0
<=> 2(x+10)(x-5)(x+10) = 0
<=> x = -10 or 5, only 5 makes sense
f"(5) = 2 + 2000/125 = 18 > 0, so x=5 is a minimum.

f(5) = 5^2 + 30(5) + 300 + 1000/5
= $675/unit.
So the minimum average cost per unit is $675, when 5 units are produced.

Answer 2

The average cost is C/x (overall cost divided by the number of items)

C/x = [(x + 10)^3]/x

To find the minimum derive it:

d/dx(C/x) = [3(x + 10)^2 * x - (x + 10)^3]/x^2

We need to find x such that
3(x + 10)^2 * x - (x + 10)^3 = 0

(x + 10)^2 * [3x - (x + 10)] = 0

(x + 10)^2*(2x - 10) = 0

Of course we won't take x=-10

2x-10 = 0 -> x=5

The average cost for x=5 is [(5 + 10)^3]/5 =
= (15^3)/5 = 15/5 * 15^2 = 3*225=675

To find if it is a minimum Let's try with an x>5 and with an x<5 (and > -10)
[Sometimes it is easier than taking a second derivative]

Take x=1
(11^3)/1 = 1331>675

Take x=10
(20^3)/10 = [20*20^2]/10 = 2*400 = 800 > 675