If $f(x)=x^7+x^5+x^3+x$, show that $f(x)=1$ has a unique solution. How to prove this

Question

What I did was

• use the intermediate value theorem for the existence of a solution
• used the first derivative to show that $f(x)$ is increasing and thus $f(x)=1$ has a unique solution

Is that correct?

Answer 1

Yes it is correct, $f'(x)>0$ implies that $f$ strictly increase, $lim_{x\rightarrow-\infty}f(x)=-\infty$, $lim_{x\rightarrow+\infty}f(x)=+\infty$ use the IVT to show the existence of an unique solution.