find the number that minimizes average cost

Question

C(x)=10x-4√(x-4)

please provide steps

Answer 1

Average cost = total cost/quantity = C(x)/x = [10x - 4sqr(x-4)]/x = 10 - 4sqr(x-4)/x

take the first derivative and set equal to zero

C'(x)/x = (-4(sqr(x-4)(-1)x^-2 + (1/x)(-1/2)(x-4)^-1/2 = 0

4sqr(x-4)/x^2 -1/2xsqr(x-4) = 0

multiply by x^2 to get

4sqr(x-4) - x/2sqr(x-4) = 0

multiply by sqr(x-4)

4(x-4) -x/2 = 0

4x-16 -x/2 = 0

7x/2= 16

x = 2/7 times 16 = 32/7 = 4 4/7

alternatively AC is when MC = AC

MC is the derivative of total cost

C'(x) = 10 - 2/sqr(x-4) = C(x)x = 10x^2 - 4x/sqr(x-4), solve for x