TITLE: Prove $x^5+10x^3+ax^2+bx+c=0$ has no more than four real roots
QUESTION [5 upvotes]: I'm trying to prove that $$f(x) = x^5+10x^3+ax^2+bx+c=0$$ can not have more than four real roots, no matter the values of $a,b,c$ real Numbers .
My attempt: $f'(x) = 5x^4+30x^2+2ax+b =0$ and $ f''(x) = 20x^3+60x+2a$, now $f(x)$ is differentiable and continuous in all real line but from here I want to use the intermediate value theorem, however I don't know how to apply it for $f$.

REPLY [10 votes]: $f'''(x)=60x^2+60$ has no real roots
$\implies f''$ has at most $1$ real root
$\implies f'$ has at most $2$ real roots
$\implies f$ has at most $3$ real roots
This argument uses Rolle's theorem at each step.