Show that $x^n = a$ has at most one real positive real if $n$ is a positive integer

Question

Show that $x^n = a$ has at most one real positive real if $n$ is a positive integer.

I can solve this question by drawing graphs for different $n$. But how should I approach the problem if I want to solve it using calculus?

Answer 1

Let $f(x)=x^n-a$. By considering the derivative, show that $f$ is increasing on $[0 , \infty]$.

So, $f(x)=0$ has at most one solution.