I need to:
Prove that if $x_0$ is the root of $a^x + b^x = c$ then no other roots exist given $a, b\in(0, 1)$ or $a,b >1$
My try is below (using monotonicity):
Case 1: $a, b > 1$
For $a, b > 1$ we know that $a^x > 0$ and is monotonically increasing, but $b^x$ has the same properties. Let:
$$g(x) = a^x + b^x >0$$ Since $g(x) > 0 \implies c>0$. The sum of monotonically increasing functions is monotonically increasing so is $g(x)$. For any $x_1 < x_2$ we have that $g(x_1) < g(x_2)$. Let $x_0$ be the root and $g(x_0) = c$. Put $x_1 < x_0 < x_2$ hence $g(x_1) < c < g(x_2)$. Given $g(x)$ is monotonically increasing there exists only one $x_0$ such that $g(x_0) = c$, which proves $x_0$ is the only root.
Case 2: $a, b \in (0, 1)$
This is different from case 1 by the fact that $a^x$ and $b^x$ are monotonically decreasing. But the same approach as above is used.
Is the proof above valid or am i missing something?