Comparing a linear function and a rational quadratic function

Question

I'm given two functions:

$$ f(x) = \frac {x^2 A + (1 - x)^2 B }{x^2 + (1-x)^2} $$ $$ g(x) = xA + (1 - x) B $$

A and B are greater than 0. In the interval $ [0,1] $, I should find when $ f(x) > g(x) $ and when $ f(x) < g(x) $.

How should I approach this problem? My first issue is that the solution will obviously depend on the value of A and B. For example, if A and B are both 1, f(x) is constant at 1. And g(x) is also constant at 1.

Answer 1

Hint: I would compute $$f(x)-g(x)?=\frac{Ax^2+(1-x)^2B}{x^2+(1-x)^2}-Ax-(1-x)B=\frac{(x-1) x (2 x-1) (B-A)}{2 x^2-2 x+1}$$