1
$\begingroup$

I have the following function of four variables

f=Sqrt[(1 + w - x + 4 y z) - Sqrt[4 (-w + x) + (1 + w - x + 4 y z)^2]]/Sqrt[2]

I want to show that $f(w,x,y,z)<1$ under the conditions $x\geq0,w\geq 1, y>0, z>0$; then, I use NMaximize[] command to find the maximum value of $f(w,x,y,z)$, that is

NMaximize[{f, x >= 0, w >= 1, y > 0, z > 0}, {w, x, y, z}]

If the maximization of $f(w,x,y,z)$ under the previous conditions gives a value less than one, I show that $f(w,x,y,z)<1$. However, the result of the previous code is

{1., {w -> 1.42498, x -> 0.00149322, y -> 0., z -> 0.0000601358}} 

Then, Mathematica carries out the numerical maximization with $y=0$ even though I'm using the condition $y>0$ in NMaximize[]. Then, my questions are:

(1) There exists some way to force Mathematica to carry out the numerical maximization with $y>0$ ?

and

(2) Is there another simpler way to show $f(w,x,y,z)<1$ (subject to $x\geq0,w\geq 1, y>0, z>0$) with Mathematica?

$\endgroup$
3
  • $\begingroup$ The expression is not always real-valued in that domain. In[189]:= f /. {w -> 1, x -> 4, y -> 1/100, z -> 100} Out[189]= I $\endgroup$ Dec 6, 2023 at 21:15
  • $\begingroup$ Evaluate Maximize[{f, x >= 0, w >= 1, y > 0, z > 0}, {w, x, y, z}] which returns Maximize::wksol: Warning: there is no maximum in the region in which the objective function is defined and the constraints are satisfied; a result on the boundary will be returned. and {1, {w -> 2, x -> 0, y -> 1, z -> 0}} This indicates that function was increasing as it approached the boundary and as you get arbitrarily close to boundary (z == 0 in this instance) the result will be arbitrarily close to 1 Look at inst = FindInstance[{f == 1 - 10^-20, x >= 0, w >= 1, y > 0, z > 0}, {w, x, y, z}, 1] $\endgroup$
    – Bob Hanlon
    Dec 6, 2023 at 21:37
  • $\begingroup$ One other way to ask: Reduce[x>=0&&w>=1&&y>0&&z>0&&Sqrt[(1+w-x+4y z)-Sqrt[4(-w+x)+(1+w-x+4y z)^2]]/Sqrt[2]>=1,{w,x,y,z}] instantly returns False $\endgroup$
    – Bill
    Dec 6, 2023 at 21:51

1 Answer 1

1
$\begingroup$

The command

FindInstance[{f >= 1, x >= 0, w >= 1, y > 0, z > 0}, {w, x, y, z}]

returns {} which, if one trusts Mathematica, proves that $f$ is strictly less than 1.

I don't personally trust Mathematica, for what it's worth.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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