Consider the function f(x,y)={xy2x2+y2if (x,y)≠(0,0)0if (x,y)=(0,0)f(x,y) = \begin{cases} \frac{xy^2}{x^2+y^2} & \text{if } (x,y) \neq (0,0) \\ 0 & \text{if } (x,y) = (0,0) \end{cases}f(x,y)={x2+y2xy20if (x,y)=(0,0)if (x,y)=(0,0) Which statement is TRUE about fff at the origin?
fff is differentiable and both ∂f∂x(0,0)=∂f∂y(0,0)=0\frac{\partial f}{\partial x}(0,0) = \frac{\partial f}{\partial y}(0,0) = 0∂x∂f(0,0)=∂y∂f(0,0)=0
Both partial derivatives exist and equal 0, but fff is not differentiable at (0,0)(0,0)(0,0)
At least one partial derivative does not exist at (0,0)(0,0)(0,0)
fff is not continuous at (0,0)(0,0)(0,0)