Answer to Question #123836 in Calculus for Pinku Kumar

Question

find fx(0,0) and fx(x,y), where (x,y) not=(0,0) for the following functi on f(x,y)={xy^3/x^2+y^2, when (x,y)not =(0,0), 0 when (x,y)=(0,0)} is fx continuous at (0,0) justify

Answer 1

  Given $f(x,y)= \begin{cases} \frac{xy^3}{x^2+y^2}, \ when \ (x,y)\neq (0,0) \\ 0 \ when \ (x,y)=(0,0) \end{cases}$

So, $f_x(0,0) = lim_{h\to 0} \frac{f(h,0)-f(0,0)}{h} = lim_{h\to 0} \frac{0-0}{h} = 0$ .

Now when $(x,y)\neq 0$ , $f_x(x,y) = \frac{(x^2+y^2)y^3 - xy^3(2x)}{(x^2+y^2)^2} = \frac{y^5 - x^2y^3}{(x^2+y^2)^2}$ = $\frac{y^3(y^2-x^2)}{(x^2+y^2)^2}$ .

Now, $f_x(x,y)$ $\to 0 \ as \ (x,y)\to (0,0)$ because numerator is more than enumerator.

So, $f_x$ is continuous at (0,0) because $lim_{(x,y)\to (0,0)} f_x(x,y) = f_x(0,0) = 0$ .