Problem

乙、計算、證明題：共 3 題，每題 12 分，共 36 分。須詳細寫出計算及證明過程，否則不予計分。

2. For the function

$$f(x,y)= \begin{cases} \dfrac{-5xy}{x^2+y^2}, & \text{if }(x,y)\ne(0,0),\\ 0, & \text{if }(x,y)=(0,0), \end{cases}$$

show that $f_x(0,0)$ and $f_y(0,0)$ both exist, but that $f$ is not differentiable at $(0,0)$.