Disproving existence of real root in some interval for a quintic equation Disprove the statement: There is a real root of equation $\frac{1}{5}x^5+\frac{2}{3}x^3+2x=0$ on the interval (1,2).
I am not sure whether to prove by counter-example or by assuming the statement is true and then proving by contradiction. 

All terms are increasing on $\Bbb R$, so the sum is also increasing. There is the trivial root $0$, hence it's the only root.

Alternately, you can factor out $x$, then you are left with a biquadratic equation $\frac15x^4+\frac23x^2+2=0$, then let $x^2=y$, and you have a trinomial to solve. But you don't even have to: it's obviously always positive.
