I have the function $f(x)={7x^6+8x+2}$ and I'm trying to prove that $f$ has exactly 2 real roots.
What I've done:
The only kind of solution I have come up with is essentially guessing pairs of values for $x$ that give $f$ a different sign and then make use of Bolzano's Theorem.
More specifically:
- $f(-1)=1>0$ and $f(-{1\over 2})=-{121\over 64}<0$, so according to Bolzano's Theorem, there is some $x \in (-1, -{1 \over 2})$ such that $f(x)=0$.
- $f(-{1\over 2})=-{121\over 64}<0$ and $f(0)=2$, so according to Bolzano's Theorem, there is some $x \in (-{1 \over 2}, 0)$ such that $f(x)=0$.
Question:
The above solution looks kind of meh to me and I don't think it proves there are exactly 2 real roots, but rather that only 2 were found. Is there a better, more convincing way to prove the existence of exactly 2 roots?