Question

Let the function,f be defined by

f(x,y) = { 3x^2y^4/(x^4+y^4) , (x,y)≠(0,0)

{ 0 ,(x,y)={0,0)

Show that f has directional derivatives in all directions at (0,0).

Answer 1

We have given the function,

$f(x,y) = \dfrac{3x^2y^4}{x^4+y^4}$ , if $(x,y) \ne (0,0)$

= 0 , $if (x,y) = (0,0)$

Direct computation yields for every $v = (x,y) \in R^2$

$D_v(f) = limt \rightarrow 0 \dfrac{f(0+tv)-f(0)}{t}$

$= 0$

Thus $f(x,y)$  has directional derivatives in all directions at $(0,0)$.