How to graphicate the function $f(x)= x^5+x+1$

Question

I know that for the zeros I have $x=-0.754888...$

That for $f'(x)=0$, I have no solutions and that $\displaystyle \lim_{x\to\infty} f'(x) = \lim_{x\to-\infty} f'(x) =-\infty$.

Using the first derivate is the only thing that occurs to me, and I can see that $f(0) = 1$, but I do not how to proceed.

Answer 1

$f'(x)=x^4+1>0$ and by your work we see that $f$ has unique real root.

Now, since $$x^5+x+1=x^5-x^2+x^2+x+1=(x^2+x+1)(x^3-x^2+1)$$ and $$x^2+x+1>0,$$ we see that the root is a root of the equation $$x^3-x^2+1=0$$ and we can get an exact value of the root by the Cardano's formula.

Also, we see the $f$ increases and $(0,1)$ is an inflection point.