Inequality In Real Number

Question

Can we find a constant $C$ independent on $a$ such that $$(a+\tau)^{\mu}-a^{\mu}\leq C \tau^{\mu},$$ where $a\in\mathbb{R}^+,\tau\in(0,1)$ and $\mu\in(1,2)$? I have found $C=1$ for $\mu\in(0,1]$, but I couldn't find $C$ when $\mu\in(1,2)$.

Answer 1

No such constant exists for any $\mu>1$. To see this, let $f(x)=x^\mu$ and observe that $f'(x)=\mu x^{\mu-1}$ goes to $\infty$ as $x\to\infty$. It follows that for fixed $\tau>0$, $f(a+\tau)-f(a)$ goes to $\infty$ as $a\to\infty$ and so cannot be bounded above.