How can I tell if one polynomial is greater than another over an interval? $y_1=ax^2 + bx + c$

$y_2=dx^2 + ex + f$

Question: what is the proper way for me to test that $y_1 &gt; y_2$ over an interval (say 0 to 10, inclusive) for all numbers in that interval?

I assume it has something to do with testing the integral of y1 - y2, but not sure.

=== Added some context ===

I'm working on a continuous game theory program written in Python, and am working on the concept of strict domination. $y1$ and $y2$ are meant to be the utility functions for a given agent. $y1$ strictly dominates $y2$ if, over the range of other player actions in question (i.e. the interval of $x$), $y1&gt;y2$.
Solve $ y_{1} = y_{2} $ and then use the solutions (if there are any) to construct the sign diagram of $ y_{1} - y_{2} $ over the interval.

Having $ \displaystyle \int_{a}^{b} (y_{1} - y_{2}) \,d{x} &gt; 0 $ is a necessary condition for $ y_{1} &gt; y_{2} $ over the interval $ [a,b] $, but definitely not sufficient.
One method is to check the first point, so in this case, check $y_1(0)\geq y_2(0)$. Then consider the derivatives and verify that $y_1'(x)\geq y_2'(x)$ for all $x\in[0,10]$.