Finding ranges of a parameter for which a function is always positive

Question

I have a complex function of a single variable expressed in analytical form, which also depends on a parameter. I would like to have Mathematica show me for which values of the parameter the real part of the second derivative of the function is always positive.

Strictly speaking, maybe I have given too much detail here, the real part/derivative stuff is ultimately inessential, what I want Mathematica to do is to tell me for what choices of a parameter a real function is positive everywhere. I hope the question is clear, though.

Answer 1

For example, Reduce:

Reduce[
 ForAll[x, α*x^2 + β*x + γ > 0],
 {α, β, γ},
 Reals
]

which looks like this:

tells us that $\alpha x^2+\beta x+\gamma>0$ for all $x$ if these conditions hold:

Or Resolve:

Resolve[
 ForAll[x, α*x^2 + β*x + γ > 0] && x ∈ Reals,
 Reals
]