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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.

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If you need a numerical approximation for a function that can't be treated analytically, you can use RegionPlot. –  Szabolcs May 24 '12 at 8:16
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1 Answer

up vote 10 down vote accepted

For example, Reduce:

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

which looks like this:

Mathematica graphics

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

Mathematica graphics

Or Resolve:

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

Mathematica graphics

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That's exactly what I was looking for! Thank you! –  zakk May 22 '12 at 22:00
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