Divide by 2
Take the first derivative
C'=2x-160 set it equal to zero 2x-160=0
divide by 2: x-80=0 or x=80
The cost function is an equation which graphed is a parabola. It's minimum point is where the slope equals zero. That's the point where the first derivative equals zero.
At x=80 cost = 6400-12800+6,010= -390 This negative cost is profit.
Or you could algebraically solve the quadratic equation a=2,
plug into the quadratic formula x=1/2a(-b + or - square root of b2-4ac)
=1/4(320 + or - square root of 3202 - 4(2)(12020)) = 1/4(320 + or - square root of 6,240)
=80 + or - square root of 6240
=80 + or - almost 79 = approximately 1 or 159
80 is the midpoint, with 1 or 159 as the break even points where cost=revenues and profit =0
80 is maximum profit or minimum cost
Plug 80 into the original cost equation, and you get the minimum cost or maximum profit.