Quadratic Function property

Question

We have: f(x) = x’Ax where A is positive definite symmetric matrix. Let’s say A is 2x2 matrix for simplicity. The intent is to find if f(x) < Climit where Climit is some positive metric. The expansion of f(x) gives f(x) = a11x1^2+2a12x1x2+a22x2^2. When I tested each element against Climit, I found that the violation comes from either a11x1^2 or a22x2^2 or both but never from 2a12x1x2 for the value of x I have tested so far. It is surprising to me that the cross term has no influence on the upper bound. Is it always the case? In other words is it true that Cross term has no influence on upper bound of f(x). If it is true, is there a mathematical proof on this observation?