0

Our math teacher told us that for rational functions (where the maximum power of x involved is 2) , we assume it equals to some variable $\lambda$ and create a quadratic in x. An example is $ y = \frac{x^2+2x+2}{x^2+x+1}$

So, $ \lambda = \frac{x^2+2x+2}{x^2+x+1}$

Rearranging we have $ x^2(1-\lambda) + x(2-\lambda) + (2-\lambda) =0$ And now apparently at the maximum and minimum values of $\lambda$ the discriminant of this quadratic equation is 0 which we use to find the range of this rational function. I do not understand why this is the case? Can we prove this?

Aspirant29
  • 1
  • Do you know how to find maxima and minima using Calculus? If so, you could see whether Calculus gives you the same answer as you get following your teacher's method. Maybe first work out some numerical examples of teacher's method, to see whether it gives sensible answers. – Gerry Myerson Oct 12 '23 at 12:29
  • Yes I know calculus. Though actually I wanted to analyze this purely by algebra to get an intuitive feeling of what exactly is happening and why the condition discriminant =0 gives us the maxima and minima which I couldn’t intuitively understand. – Aspirant29 Oct 13 '23 at 06:55
  • Well, as I suggested, before spending any time trying to prove it, you should run some experiments to see whether it gives the right answers, by comparing what it gives to what Calculus gives. – Gerry Myerson Oct 13 '23 at 12:09
  • See also https://math.stackexchange.com/questions/1768078/why-d%e2%89%a50-while-finding-the-range-of-rational-functions – Gerry Myerson Oct 16 '23 at 04:49

0 Answers0