Prove that a degree-$6$ polynomial has exactly $2$ real roots

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?

Hint: by Descartes' rule of signs the equation has no positive real roots, and at most $2$ negative ones. But you showed that it has at least one real root (and it's enough that $\,f(-1/2) \lt 0 \lt f(0)\,$ for that), then it must have a second real one, since non-real complex roots come in conjugate pairs.

$$f''(x)=7\cdot6\cdot5x^4\geq0,$$ which says that $f$ is a convex function.

Thus, $f$ has two roots maximum and by your work we get two roots exactly.

Hint

You proved so far that there are at least 2 solutions.

If the function would have 3 or more solutions, then by Rolle's Theorem, $f'$ would have at least 2 solutions.

We have:

$f(x)=7x^6+8x+2$. On differentiating,

$f'(x)=42x^5+8$ which has one solution and the solution is $x=-0.718$. Now, you can be sure that your polynomial equation has at the most two roots. To prove exactly two roots, divide the real line in two intervals $(-\infty,-0.718]$ and $[-0.718,\infty)$. Now, you can check (using the sign of derivative) in the first Interval, the function is decreasing and in the second Interval function is increasing.

Also, note that the function takes positive values at the end of Interval and negative value in the neighborhood of your critical point which is $-0.718$. Can you guess what this means. This means, the function cuts the x-axis two times. In the first and second Interval. Hence, function has exactly two roots.