How do I find all values of mu and k such that my function is nonnegative?

Question

I have this nasty function in terms of $\mu$, $k$, $x$, and $z$ that I want to study when it's nonnegative (part of my research project):

$$ -9 \sqrt{2 \pi } k^{5/2} z (x-\mu )^4 e^{\frac{1}{2} k (x-\mu )^2}-4 \sqrt{2 \pi } k^{3/2} z (x-\mu )^2 e^{\frac{1}{2} k (x-\mu )^2}+6 \pi k^3 z^2 (x-\mu )^4 e^{k (x-\mu )^2}+6 k^2 (x-\mu )^4+k \left(-6 \pi z^2 e^{k (x-\mu )^2}+4 \mu ^2+4 x^2-8 \mu x\right)+7 \sqrt{2 \pi } \sqrt{k} z e^{\frac{1}{2} k (x-\mu )^2}-4 \geq 0$$

How would I code up a script to determine values $\mu, k$ such that the above holds for all $x$ such that $|x-\mu|<\epsilon$ and $z\in \mathbb{R}$. I'm just having trouble because I tried to run

Minimize[-4 + 7 E^(1/2 k (x - \[Mu])^2) Sqrt[k] Sqrt[2 \[Pi]] z - 
  4 E^(1/2 k (x - \[Mu])^2) k^(3/2) Sqrt[2 \[Pi]] z (x - \[Mu])^2 + 
  6 k^2 (x - \[Mu])^4 - 
  9 E^(1/2 k (x - \[Mu])^2) k^(5/2) Sqrt[2 \[Pi]] z (x - \[Mu])^4 + 
  6 E^(k (x - \[Mu])^2) k^3 \[Pi] z^2 (x - \[Mu])^4 + 
  k (4 x^2 - 6 E^(k (x - \[Mu])^2) \[Pi] z^2 - 8 x \[Mu] + 
     4 \[Mu]^2), {\[Mu], k}]

but that didn't work at all. I also need to add in the above constraint on $x$.

Any advice or help is appreciated.

Answer 1

Not a complete answer but these points maybe of some help to you:

• You can use FullSimplify on your equation to return the following:

$$2 k (x-\mu )^2 \left(3 k (x-\mu )^2+2\right)+6 \pi k z^2 e^{k (x-\mu )^2} \left(k (x-\mu )^2-1\right) \left(k (x-\mu )^2+1\right)-\sqrt{2 \pi } \sqrt{k} z e^{\frac{1}{2} k (x-\mu )^2} \left(k (x-\mu )^2 \left(9 k (x-\mu )^2+4\right)-7\right)-4$$

• Since there is a $\sqrt{k}$ involved, the region is restricted to $k \geq 0$.
• See that the whole equation is symmetric about $(x-\mu)$. So you can essentially fix a $\mu$ and only worry about the region $(-\mu, \infty)$.
• The only region of negative values could be due to absence or low contribution from $e^{(x-\mu)^2}$, so near low values of $x$.
• The only parameter which does not seem to have any such constraints is $z$.
• The rest can be found out by using

With[{\[Mu] = 0}, 
 RegionPlot3D[-4 + 6 E^(k (x - \[Mu])^2)
      k \[Pi] z^2 (-1 + k (x - \[Mu])^2) (1 + k (x - \[Mu])^2) - 
      E^(1/2 k (x - \[Mu])^2) Sqrt[k] Sqrt[2 \[Pi]]
      z (-7 + k (4 + 9 k (x - \[Mu])^2) (x - \[Mu])^2) + 
    2 k (2 + 3 k (x - \[Mu])^2) (x - \[Mu])^2 >= 0, 
   {x, 0, 100}, {z, -100, 100}, {k, 0, 10}, AxesLabel -> Automatic, 
  ImageSize -> Large, ViewPoint -> {-2, -2, -2}, 
  Lighting -> {{"Ambient", LightBlue}}]]

• So do not use low values of $x$ and $k$. You can use Minimize also.