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 to get the answer. 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