Originally Posted by

**transgalactic** $\displaystyle

x^{13}+x^7-x-2006=0\\

$

prove that there is a solution on this interval

$\displaystyle

\left ( \sqrt[6]{\frac{\sqrt{101}-7}{26}},0\right )

$

i tried like this

f(x)=x^13 +x^7 -x -2006

f'(x)=13x^2+7x^6 -1

f''(x)=156x^11+42x^5

t=x^6

the extreme points are

minimum point is $\displaystyle x1= \sqrt[6]{\frac{\sqrt{101}-7}{26}}=0.699$

maximum point is $\displaystyle x2= -\sqrt[6]{\frac{\sqrt{101}-7}{26}}=-0.669$

f(x1) and f(x2) are negative

then they just input 10 and its positive

so we have one solution between x=0.699 and x=10 (the mid something theorem)

but its far away from the asked range

??