Marginal and Total Revenue: A Mathematical Link

For many, the concept of the marginal is a tough one. I've always enjoyed economics with a mathematical edge, and this is quite a quaint way of tying the two incredibly linked economic ideas together.
We know the relationship that when MR=0, TR is maximised (as once MR becomes negative TR must fall). But this relationship can be explained using first differentials and turning points.
We know that MR is the gradient function of TR, and as this value is always falling (negative bell shape) we know that MR must have a negative gradient. Also, the turning point of the TR curve is where the gradient function has a value of zero, or essentially:
QTRmax = QMR0.


If you look at a total revenue curve, its upside-down bell curve shape resembles that of a negative quadratic, and that is essentially what it is. Therefore, we can give this TR curve a general mathematical formula of y=ax-x^2 (where a is a positive integer).
Because of the nature of the MR curve, we can differentiate the TR curve to find a generic formula for this curve as well:
y = ax - x^2
dy/dx = a - 2x
This formula matches the negative linear function of the MR curve, hence when MR=0:
a - 2x = 0
a = 2x
a/2 = x
Therefore revenue maximisation is achieved where x (or quantity) is equal to half the distance between the two x axis intercepts of the curve.
Now we didn't necessarily need to differentiate to show that a/2 is the output where revenue is maximised, as we can see it by symmetry of the TR curve, but the use of the maths just proves it nicely.
This esssentially proves that the value of a/2 is the output where revenue is maximised, so for those economists who struggle with the wordy explanations of the relationship, this nice link may be helpful.
From this we can find the TR at any value of x, merely by setting the value of a and hence x for a specific firm.