Re: Root again

• To: mathgroup at smc.vnet.net
• Subject: [mg109029] Re: Root again
• From: Bill Rowe <readnews at sbcglobal.net>
• Date: Sun, 11 Apr 2010 04:30:20 -0400 (EDT)

```On 4/10/10 at 6:52 AM, petsie at dordos.net (Peter Pein) wrote:

>while experimenting further with Root[1 + t*#1 + #1^6 &, k], k=1..6,
>k element N

>I met the following issue:

>1.) I want to know for which values of t>0 the Roots are
>real-valued:

>In[4]:= (NMinimize[{t, Im[#1] == 0 && Im[t] == 0}, t] & ) /@
>Table[Root[1 + t*#1 + #1^6 & , k], {k, 6}]

<error messages snipped>

I really don't understand why you would attempt to determine the
answer using NMinimize. It seems to me the first step is a
simple plot of the roots as a function of t. That is

In[18]:= r = Table[Root[1 + t*#1 + #1^6 &, k], {k, 6}];

In[19]:= Plot[r, {t, 0, 10}]

The resulting plot indicates half the roots are increasing with
increasing t and the other half are decreasing with increasing
t. This alone means there will be problems with using NMinimize

Looking at the first root object I will use Maximize since the
plot indicates this root is decreasing with increasing t

In[20]:= sol = Maximize[{r[[1]], t > 0}, t]

Out[20]= {-Root[5 #1^6-1&,2],{t->Root[3125 #1^6-46656&,2]}}

In[21]:= sol[[1]] // N

Out[21]= -0.764724

In[22]:= N[t /. sol[[2]]]

Out[22]= 1.56919

So, it would seem the roots are all real for any t greater than 1.56919

```

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