ATC=C(Q)/Q=100/Q+2+3Q
For minimization differentiate w.r.t. Q and equate to zero the equation is -100/Q^2+3=0 and the answer is Q=5.77.
Since Q can take only whole units, calculating ATC at Q=5 comes out to C(Q=5)=100/5+2+3*5=37. When ATC is calculated at Q=6, it comes out to C(Q=6)=100/6+2+3*6=36.67. Since cost is minimum at Q=6, the ATC is minimized at Q=6. What is the total cost of an output of 500?