Suppose the cost function is C(Q)=50+Q-10Q22+2Q33

The cost function is given as follows:

(Q)=50+Q−10Q2+2Q3(Q)=50+Q−10Q2+2Q3

ATC of the cost function is calculated as follows:

(Q)Q=50+Q−10Q2+2Q3Q=50Q+QQ−10Q2Q+2Q3Q=50Q+1−10Q+2Q2(Q)Q=50+Q−10Q2+2Q3Q=50Q+QQ−10Q2Q+2Q3Q=50Q+1−10Q+2Q2

To know that a function is in increasing stage or decreasing stage, the first derivative of ATC is required because the first derivative shows the rate of change of a function, which is calculated as follows:

dATCdQ=−50Q2−10+4QdATCdQ=−50Q2−10+4Q

At Q equal to 10, the value of the first derivative of ATC will be:

ATC′10=−50102−10+4×10=−50100−10+40=−12+30=2912ATC10′=−50102−10+4×10=−50100−10+40=−12+30=2912

Hence, the value of first derivative of ATC is greater than 0 at Q equal to 10, which shows that the ATC is in the increasing stage.

ATC cannot be in the decreasing stage because value of first derivative of ATC is greater than 0.

The ATC will not be at its minimum level as the first derivative of ATC is not equal to zero. The ATC is increasing in this case.

ATC is not at the minimum level because the value of first derivative of ATC at Q equal to 10 is greater than 0.

To be at a minimum level, the value of first derivative of ATC should be 0 at Q equal to 10, and the value of the second derivative of the ATC should be positive.

Hence, ATC is not at the minimum level.