5
$\begingroup$

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.

$\endgroup$
  • 1
    $\begingroup$ If you need a numerical approximation for a function that can't be treated analytically, you can use RegionPlot. $\endgroup$ – Szabolcs May 24 '12 at 8:16
11
$\begingroup$

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

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.