Prove that the polynomial has exactly 2 real roots by IVT or Rolle's Theorem

Question

I'm trying to prove this by IVT or Rolle's Theorem. Usually if it would say "Prove it has only 1 real root", I would assume it had 2 roots, take the derivative and if it was $≠0$, then I would prove that it has only 1 root, so Rolle's Theorem.

Im kind of stuck on this one, because I'm not able to suppose anything and then give a counterexample:

$6x^4-7x+1=0$

Any tip or help would be much appreciated.

Answer 1

Hint: Show that the minimum value of the function $f(x) = 6x^4 - 7x + 1$ is negative. If $a$ is the value of $x$ such that $f(a)$ is the minimum value of $f$, then show that $f$ is decreasing over all $x < a$ and increasing over all $x > a$.