In the long run the cost of 100 is quasi-fixed, meaning that it can be escaped by choosing q = 0

(a)

C = q² + 10q + 100

AC = (C/q)

=> AC = q + 10 + (100 /q)

and

MC = dC / dq

=> MC = 2q + 10

The minimum value of Average cost is the minimum price at which a perfectly competitive supply in long run.

Note: Long run equilibrium point is P = MC = AC

The average cost is minimum at MC = AC

=> MC = AC

=> 2q + 10 = q + 10 + (100 /q)

=> 2q + 10 - q - 10 = (100 /q)

=> q = (100 /q)

=> q² = 100

=> q = (100)^1/2

=> q = 10

At q =10, the AC is minimum.

AC = q + 10 + (100/q)

=> AC = 10 + 10 + (100/10)

=> AC = 10 + 10 +10

=> AC = 30

The minimum value of AC is 30.

Hence, in the long run the minimum price at which this firm would supply a positive output is 30 and at this price it supplies an output of 10 units.

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(b)

In the long run, if price is 20, then firm will supplies an output of zero units, because Price is less than 30 (i.e., the minimum price of long run)

At this price the profit will be zero because the TR and TC both will be zero