Let f: R^2 - { (0, 0)) rightarrow R be defined by f(z, y) = (2x + y)^2/x^2 + y^2
ID: 3111821 • Letter: L
Question
Let f: R^2 - { (0, 0)) rightarrow R be defined by f(z, y) = (2x + y)^2/x^2 + y^2. a) Can f be defined at (0, 0) so that f is continuous there? That is, does there exist a real number a such that setting f(0, 0) = a means that f is defined on all of R^2 and f is continuous at (0, 0)? If so, what is a? If not, why not? b) Can f be defined at (0.0) so that partial differential f/ partial differential x(0, 0)exists? That is, does there exist a real a real number b such that setting f(0, 0) = b means that partial differential f/ partial differential x(0, 0) exists? If so, find b and compute partial differential f/partial differential x(0, 0). If not, why not? c) Can f be defined at (0, 0) so that partial differential f/partial differential y(0, 0) exists? That is, does there exists a real number c such that setting f(0, 0) = c means that partial differential f/partial differential y(0, 0) exists? If so, find c and compute partial differential f/partial differential y(0, 0). If not, why not? d) Can f be defined at (0, 0) so that both partial differential f/partial differential x(0.0) and partial differential f/partial differential y(0.0) exist? If so, define f appropriately at (0, 0) and find both partials. If not, why not?Explanation / Answer
To test continuity take the path test since x tends to zero and y tends to zero simultaneously at 0,0 substitute y=mx with x tending to zero.now you solve the function you will get (2+m)^2/1+m^2 which is dependent on m and hence the function is discontinuous and there is no a
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