Lipschitz constant $k$ with respect to $y$ for $f(t,y) = te^{-y^2}$

Question

Find a Lipschitz constant $k$ with respect to $y$ for $f(t,y) = te^{-y^2}$, $\{(t,y): |t| \leq1, y \in \mathbb R\}.$

Attempt :

$|f(t,y_2) - f(t,y_1)| = | te^{-y_2^2} - te^{-y_2^2} | = |t(e^{-y_2^2} -e^{-y_1^2})| = |t||e^{-y_2^2} -e^{-y_1^2}| $

We have that : $|t| \leq 1 \Leftrightarrow |t||e^{-y_2^2} -e^{-y_1^2}| \leq |e^{-y_2^2} -e^{-y_1^2}|$, since $|e^{-y_2^2} -e^{-y_1^2}| > 0$

which gives me a constant $M=|e^{-y_2^2} -e^{-y_1^2}|$ but isn't with respect to $y$.

Now, I know that you generally need to set an inequality to proceed, but I seem to be stuck on finding another one. Also, I've taken some $y_1,y_2 \in \mathbb R$ which I'm not sure that I'll be able to get a constant $k$ with repsect to $y$ out of them, as mentioned above.

I'm new to this kind of problems, so please, any thorough solution and explanation would be much appreciated. Thanks in advance !

Answer 1

Hint: the standard trick is the Mean Value Theorem: $$|e^{-y_2^2} -e^{-y_1^2}| = |\frac{d}{dy}e^{-y^2}|_{y=y_0}||y_1 - y_2|$$ for some $y_0$ between $y_1$ and $y_2$.

You know how to bound $|\frac{d}{dy}e^{-y^2}|_{y=y_0}|$?