Given, vector-valued function $\phi(x)=\left[ \begin{array} aa x_{1}^{2}/(x_{1}+b)+x_{1}x_{3}+\frac{6}{\pi}arctan(\frac{x_{1}}{b}-180)+e+c \\ x_{1}x_{3}+x_{2}x_{4}\\ 0\\ 0\\ 0\\ 0\\ 0 \end{array}\right]$ where $x=[x_{i}]^{T},i=1,...,6$ and $a,b,c,e$ are known positive constants. All the $x_{i},i=1,...,6$ are bounded and positive.
Q1/ Does this $\phi(x)$ satisfies the Lipschitz nonlinearity condition locally or globally? If yes, then kindly help me with the proof.
Q2/ Calculate the Lipschitz constant?
1) Wherever the function is continuously differentiable, it is Lipschitz.
2) Lipschitz implies continuous everywhere and differentiable almost everywhere, so consider the largest subset $S$ of the domain which satisfies those two conditions. Wherever the function is differentiable in $S$, if the derivative is bounded, then it is Lipschitz.
In either case, the constant is $L = \sup\limits_{x\in S} |f'(x)|$.
These two conditions will usually allow you to determine if a function given in closed form is Lipschitz. When dealing with a vector valued function, you can just look componentwise, and make sure you're using the correct definition of differentiable for multivariable functions (not just existence of first partials).
These conditions will probably not help if you have a nonstandard function defined by integral, power series, or as the solution of a given equation, so it's not flawless.