The firm's average total cost curve is:

TC(Q)=a+bQ+cQ2+dQ3TC(Q)=a+bQ+cQ2+dQ3

If a=50,b=12,c=−1, and d=0.1a=50,b=12,c=−1, and d=0.1, the total cost becomes:

TC(Q)=50+12Q−Q2+0.1Q3TC(Q)=50+12Q−Q2+0.1Q3

The shutdown point is defined as the point where P=MC=AVCP=MC=AVC.

From the total cost curve above, the marginal cost is:

MC=ΔTC(Q)ΔQ=12−2Q+0.3Q2MC=ΔTC(Q)ΔQ=12−2Q+0.3Q2

The variable cost curve is:

VC(Q)=12Q−Q2+0.1Q3VC(Q)=12Q−Q2+0.1Q3

And the average variable cost is:

AVC=VCQ=12−Q+0.1Q2AVC=VCQ=12−Q+0.1Q2

Equating the average variable cost to the marginal cost, we get:

12−2Q+0.3Q2=12−Q+0.1Q20.2Q2=Q00.2Q=10Q∗=512−2Q+0.3Q2=12−Q+0.1Q20.2Q2=Q00.2Q=10Q∗=5

Therefore, the firm's shutdown price is:

P=MCP=12−2Q+0.3Q2P=12−2(5)+0.3(5)2P=$9.5