Lipschitz Question Determine if following function satisfiy Lipschitz or not 1)

Question

Lipschitz Question Determine if following function satisfiy Lipschitz or not 1) f(x,y) = (2x + 2y -3)^2/3 for |x| <=1 , |y|<=1
2) x^2cos^2y + ysin^2x for |x|<=1, |y| <infinity Lipschitz Question Determine if following function satisfiy Lipschitz or not 1) f(x,y) = (2x + 2y -3)^2/3 for |x| <=1 , |y|<=1
2) x^2cos^2y + ysin^2x for |x|<=1, |y| <infinity Lipschitz Question Determine if following function satisfiy Lipschitz or not 1) f(x,y) = (2x + 2y -3)^2/3 for |x| <=1 , |y|<=1
2) x^2cos^2y + ysin^2x for |x|<=1, |y| <infinity

Explanation / Answer

1.) f(x,y) = (2x + 2y -3)^2/3

Computing the partial derivative of y with respect to x,

del f/ del x = 2/3(2x+2y-3)^(-1/3)*2

del f/ del x = 4/3 (2x+2y-3)^(-1/3)

As we know |x| and |y| <=1.

So, del f/del x <= 4/3

So, This function satisfies Lipschitz function.

2.) Similarly for this part.

Let find del f/ del x,

del f/ del x = 2x cos^2y +y*2sin x cos x

So, del f/ del x = 2x cos^2y + y sin 2x

for |x| <= 1 and |y| < infinity

del f/ del x is infinite

So, it is not satisfying Lipschitz.

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