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Q about Interval arithmetic

In the following lines Interval arithmetic gives an Interval that
contains  the range of poly[x] for the domain = Interval[{ -1.0, 0.5

poly[x_] := 15*x^7 + 15*x^2 - 13*x + 15;

poly[ Interval [{ -1.0, 0.5 }] ]

Interval[{-6.5, 43.1172}]

Roots[eqn, vars] can not find the roots of  poly'[x]  in closed form. 
This  is probably because it isn't possible to do so.  Hence I would
figure it's  not possible to determine the exact range of poly[x] over
the given domain. 
 However, the exact range is a subset of the Interval in Out[2]. As I
understand it this should always be the case.

In the lines below I used Plot and FindMimimum to convince myself that a
smaller Interval containing the exact range and nearly equal to the
exact  range is:
Interval[{ 12.221, 32.082 }]

Is it possible to improve on the built in algorithm (developed by
Wolfram  Research)?
Is there a known algorithm that can be used to obtain the smaller ( 
preferred ) Interval?
I am not looking for an approach that uses Numerical methods as I did
below. I want is something that provides gaurenteed results, and works
on high  order polynomials.



{12.2202, {x -> 0.4154}}

FindMinimum[ -poly[x], {x,-0.8} ] //MapAt[ Minus, # ,1]&

{32.0813, {x -> -0.84552}}

     Ted Ersek

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