2
$\begingroup$

For what values of $\alpha$ and $\beta$ the derivative of the function is bounded?

Let $f_{\alpha,\beta}(t)= \left\{ \begin{array}{lcc} t^\alpha \sin(t^\beta) & si & 0 < t \leq 1 \\ \\ 0 & si & t = 0 \end{array} \right.$

Then $$|f'_{\alpha,\beta}(t)|=|\alpha t^{\alpha-1}\sin(t^\beta)+\beta t^{\alpha + \beta-1 }\cos(t^\beta)|\leq|\alpha t^{\alpha-1}\sin(t^\beta)| +|\beta t^{\alpha + \beta-1 }\cos(t^\beta)| $$

I know that $|\sin(t^\beta)| \leq 1 $ and $|\cos(t^\beta)| \leq 1$ but here i'm stuck. can someone help me?

$\endgroup$

1 Answer 1

0
$\begingroup$

To start you off: if $\alpha > 1$ and $\beta\ge 0$, both terms in $f'_{\alpha,\beta}(t)$ go to $0$ as $t \to 0$. Now consider the other cases...

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .