Hello Christy.
This question asks (implicitly) to find the maximum profit, which in this case is the point where you create the maximum number of dresses.
Think of the function as a curve on a regular plane (graph) of two variables, L along the horizontal axis and Q along the vertical axis. It has some shape. If you pick a point on the curve and look at the slope of the curve at that point, it will be negative, positive or zero. When it is negative, then moving to the left is a higher point on the curve (a larger Q). If the slope is positive, then moving to the right is a higher point on the curve (a larger Q). If the slope is zero, then the point is either a minimum or a maximum, depending on whether the curve at that point is concave up (then it is a minimum) or concave down (then it is a maximum).
So you want to find the critical points of the function Q = L  L^{2}/800.
To do this, you find the derivative which tells you the slope for some value L. I'll make a (wrong) derivation and say the derivative is dQ/dL = 2  L/1000 (I want you to do the derivation yourself with the right values, but I want to show you how to solve the problem generally).
Then the critical points are where dQ/dL = 0 = 2  L/1000 which is just L = 2000. Since that is the only value of L that makes dQ/dL equal zero, that's probably as much as we need to solve, but just to be sure, we should use the Second Derivative Test to check the concavity and therefore whether it is a minimum or a maximum.
To get the second derivative, differentiate again: d^{2}Q/dL^{2} = 1/1000. All we are interested in is whether this is negative or positive at the critical value(s) of L. Clearly, it is negative always. That means the curve is concave down at this point which tells us that this critical point is a maximum, as we'd hoped.
Now we know what value of L makes Q its maximum. So we know the number of hours of labour to hire and therefore the cost of the labour. If we calculate the corresponding Q for this value of L (just put it into the original function), we know the number of dresses manufactured. Multiply this by the price per dress and subtract the cost of the labour and you get your profit. Note that the question assumes that there are no other costs (like material or equipment), so the profit calculation is easy once you have calculated L.
Hope this helps,
Stephen La Rocque.
