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. 

First step: what is the value of the polynomial for $x=1$?
Second step: what is the sign of the derivative of this polynomial on the interval $[1,2]$? 
Third step: conclusion.
edit
Altenative: $x\ge 1$, therefore $x^n\ge 1$ for all $n\ge 1$. Hence $$\frac{1}{5}x^5+\frac{2}{3}x^3+2x\ge \frac{1}{5} +\frac{2}{3} +2 $$
whenever $x\ge 1$. 
