Lipschitz constant for $f(x,y)=e^x \sin y$

Question

Calculate Lipschitz constant for $$f(x,y)=e^x \sin y$$ on $R=\{|x|\leq 1, |y|\leq 2\pi\}$

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My finally answer is $L=e=2.718$

$L$ is Lipschitz constant

Because

$|f(x,y_2)-f(x,y_1)|=|e^x(\sin y_2-\sin y_1)|$

$=|e^x|.|\sin y_2-\sin y_1|$

$\leq |e^x| |y_2-y_1|$

Because [$|\sin a - \sin b| \leq |a-b|$]

Is the last inequality true ?

$|f(x,y_2)-f(x,y_1)|\leq |e^1||y_2-y_1|$

Then $L=e$

True?

Answer 1

You're correct, but another way to do this is to notice that the function $f(x,y)$ is smooth. So with that, we can simply maximize the gradient of $f(x,y)$ over the domain you want and this will be our Lipschitz constant.