Proving equation has solution for every $c ≥ 0$

Question

Task: Proof that the equation $x^5 − x = c$ has a solution for every $c \ge 0$ in the interval $[0, \infty)$.

No idea where to start, anyone have any suggestions?

Kind regards

Anthony

Answer 1

HINT

Fundamental Theorem of algebra implies the polynomial $x^5-x-c$ has $5$ roots, of which there must be an even number of complex roots, so there must be at least one real root.

Answer 2

$x^5-x$ is $0$ when $x=1$

and continuous and increasing (since the derivative $5x^4-1\ge0)$ without bound for $x\ge 1$,

so, for any $c\ge0$, there is $x\ge0$ (in fact $x\ge1$) such that $x^5-x=c$.