Question

Examine whether the second order partial derivatives of f at (0,0) exist or not if

f :R² ➡R is defined by

f(x, y) ={x²y/√x+y² , xy≠0 and 0, xy=0

Answer 1

$\text{By definition }f:\R\rightarrow\R,\text{defined by,}\\ f(x,y)= \begin{cases} \frac{x^2y}{\sqrt{x+y^2}} & \text{if}\ (x,y)\neq(0,0)\\0 & \text{if\ (x,y)}=(0,0) \end{cases}\\ \text{Now,}\\ f(x,0)=f(0,y)=0\\ \text{Thus, }f_x(x,0)=f_y(0,y)=0\\ \Rightarrow f_{xx}(x,0)=f_{yy}(0,y)=0\\ \text{Thus, the second partial derivatives of the given function exists at (0,0) and are: }\\ f_{xx}(0,0)=f_{yy}(0,0)=f_{xy}(0,0)=0$