Question
Check whether satisfies Lipschitz condition
i) on any rectangle and
;
ii) on any strip and
;
iii) on the entire plane.
Answer :
Word Count : 372
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Skip in 10
We are asked to check whether ( f(x, y) = x^2 y^2 ) satisfies the Lipschitz condition in three domains. A function ( f(x, y) ) is Lipschitz continuous in (y) if there exists a constant ( L ) such that [ |f(x, y_1) - f(x, y_2)| \le L |y_1 - y_2| ] for all ( x ) and ( y_1, y_2 ) in the given domain. Similarly, for Lipschitz in (x), we check differences in (x). The partial derivative with respect to ( y ) is [ \frac{\partial f}{\partial y} = 2x^2 y. ] The magnitude is ( | \frac{\partial f}{\partial y} | = 2 |x^2 y| ). --- i) On a rectangle ( a \le x \le b, \ c \le y \le d
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