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Problem 4 (due Monday, March 25)
A function f:R2⟶R has the following properties:
a) the partial derivatives ∂f∂x and ∂f∂y are continuous on R2;
b) (∂f∂x(x,y))2+(∂f∂y(x,y))2≤∂f∂x(x,y) for every (x,y)∈R2;
c) f(x,0)=0 for all x∈R.
Prove that f(x,y)=0 for all (x,y)∈R2.
We received only one (partial) solution, from Beatrice Antoinette. For a complete solution see the following link Solution.