Total cost function and (dis)economies of scale I'm really confused about how I'd tackle this. 

I have an equation, $TC(Q) = 100Q + 20Q^2 + 3Q^3$.

And I'm trying to find where the economies of scale, diseconomies of scale and constant return to scale occurs.

My instinct would be to take the derivative of that function, and then set it to 0, and then see what the values to the left and right of 0 are which would be the slope and allow me to see depending on whether or not they're positive or negative whether it's an economy or diseconomy of scale.

Is this along the right road?
First of all, knowing the cost function is not enough to know whether we have decreasing, increasing or constant returns to scale. Returns to scale are a property of the firm's production function -- and the relationship of the production function to the cost function is not as straightforward as you might think. However, if we assume that the firm takes factor prices as given, then we can apply a (rather intuitive) result from Sandmo (1970):


  Average cost is decreasing, constant or increasing according as returns to scale are increasing, constant or decreasing.


In light of this, we simply need to consider the average cost curve:

$$ AC(Q) =100+20Q+3Q^2 $$

Clearly, this is strictly increasing in $Q$. Therefore, if we are willing to assume price taking in the factor markets, we know that the firm's underlying production function exhibits decreasing returns to scale.