 Hud also applied his rule to finding extreme values of rational functions. The rule is actually more complicated to describe than it is to apply. So we'll look at some examples. So let's say we want to find the extreme values of this rational expression. As before, we'll assume that the expression has some maximum value z, and it's a rational expression, so we can do a little bit of algebra and convert it into a polynomial. So applying Hud's rule, we'll select an arithmetic sequence and multiply the terms of the sequence, and it's convenient if our sequence ends in zero, which will eliminate one occurrence of our unknown z, and we need the roots of this new polynomial. Now, while it's obvious that x equals zero is a root, it can't be a double root of our original equation, and so we can remove a factor of x, and we still have two unknowns. We don't know x and we don't know z. If only we had some expression that told us what z was equal to. Oh, wait, we do. And so we note that z is equal to the rational expression itself, so we can write down our equation, replace, and solve, and this gives us the candidates for our repeated roots. So let's think about that. x is equal to five, our z value is equal to nine, and we want to find the solution to, and we confirm that x equals five is a double root, and so this is going to correspond to one of our extreme values. Likewise, if x equals negative one, then our z value is, and solving our equation gives us, and we see that x equals negative one is a double root, and so our repeated roots are five and negative one, and through other means, we find that x equals five corresponds to a local minimum, and x equals negative one to a local maximum. Hood also showed how his method could be used to find what we'd call constrained optimization problems. This is maximize some expression, l subject to the constraint that x and y satisfy a given relationship. Hood tackled the problem of finding the widest point on the folium, but this is very challenging, so we'll look at a simpler problem that relies on the same ideas. So let's try to find the maximum value of x, y on the ellipse, x squared plus four y squared equals 50. So we'll let the maximum value be x, y equals z, then y is z over x, and we can substitute this into our equation, and so what we want is we want this equation to have a double root. So multiplying by an arithmetic sequence setting in zero, and note that we do have to take into account that our x cubed and x terms have coefficient zero, and we find the roots, and obtain zero, five, and negative five, what we might call our critical values. So if x equals zero, z is equal to zero, and our equation becomes, we see that x equals zero will be a double root, so it might correspond to an extreme value, but note that if x equals zero, x, y is zero, so we won't get a maximum or a minimum. If x equals five, then we find that y is equal to five halves, so z is 25 fourths, and again we want to check to see if x equals five is a double root of our equation, and we see that both x equals five and x equals negative five are double roots, and so these correspond to our maximum values. Hud's rules made the algebraic calculus affirm a Descartes practical because they could be applied to any algebraic relationship between x and y. This meant you could find the tangent to the folium, x cubed plus y cubed equals pxy, the extreme values of some completely arbitrary algebraic function or the maximum value of any algebraic expression along any algebraic curve. We'll leave the details to the viewer, and Hud's two letters established him as one of the rising stars of mathematics. Unfortunately, Hud chose to abandon a promising career as a mathematician and went into politics.