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

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.

• Might be more tractable if you use FullSimplify and then replace x-\[Mu] with a new variable. – Daniel Lichtblau Apr 4 '20 at 17:10

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

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

• 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.