The value of k above which the function f where f(x,y)=x+y|+(x+y)is everwher differentiable

Ashwini I suggest you to take this question on hold. Because there is a loophole on the points lie on the line y=-x by both way hit and trial and piecewise defined fn as above.

Sir, I think piecewise defined fn in last step is not even continuous on points that lies on line y=-x.

K=0

I am just telling you that this hit and trial method sometimes work in exam too in NAT section. But at this time you have time so do this question with right method.

If we do not talk about method and just simply observe, Then we will get that if we take k=0 then given function becomes constant function that is 1, which is differentiable everyw...

we can take value k=0, it will satisfy given statement.