Find a Lipschitz constant for the function $f(t,x)=x(\sin t -x^2)$

Question

The function is bounded on t from $0$ to $2\pi$ And $\vert x \vert \le M$

See that $\vert f(t,x)-f(t,x')\vert \le L\vert x-x'\vert$

I have taken the derivative with respect to x, but don't know what to do from here, does it suffice to say that the Lipschitz constant will be an arbitrary M^3?

Any help regarding multivariate Lipschitz functions would be appreciated

Answer 1

we have $$f(t,x_2)-f(t,x_1)=x_2(\sin(t)-x_2^2)-x_1(\sin(t)-x_1^2)=\sin(t)(x_2-x_1)-(x_2^3-x_1^3)=\sin(t)(x_2-x_1)-(x_2-x_1)(x_2^2+x_1x_2+x_1^2)$$ does this help you? thus we have $$|f(t,x_2)-f(t,x_1)|\le |x_2-x_1|\left(\left|\sin(t)\right|+|x_2^2+x_1x_2+x_1^2|\right)$$