Suppose there is a function $f(x)=a_nx^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\cdots +a_1x+a_0$ and it is given that $a_n>0$ and $f(x)>0\:\: \forall x\in \mathbb{R}$ Then will it be accurate to say that the $\Delta$ of $f(x)$ will always be $<0$$?$
There was a question this which led me to this feeling of uncertainty.
Because, as far as I know the theory of
$\Delta \lt0$ if $f(x)\gt0$ (assuming that the leading coefficient is positive) is only applicable when $f(x)$ is a quadratic polynomial.
Or am I wrong$?$ Any ideas will be greatly appreciated.
As pointed in the comments, for higher degree polynomials discriminant does not carries all the information.Discriminant becomes more and more complex as you go higher and higher. And you should try to find some counterexamples (if they exist) .