2

Given the function $$ f(x,y)= \begin{cases} \frac{\sin(x)^4 \ln(1+x^2)}{(1+\cos(x))^2+y^4}, & \text{if $(x,y)\neq (0,0)$} \\ 0, & \text{if $(x,y)=(0,0)$} \end{cases}$$ I want to check if it is differentiable at $(0,0)$.
First I checked if it is continuous at $(0,0)$, I saw it is.
Then I tried using the definition of differentiable function but couldn't get a definite result.
Any ideas?

Chiranjeev_Kumar
  • 3,061
  • 16
  • 30
Yinon Eliraz
  • 875
  • Have you found the partials are the origin? Do you know how this helps? – Git Gud Jul 12 '15 at 17:26
  • 3
    Every function of $x$ and every function of $y$ involved is $C^\infty$ at $0$ and the denominator is non zero in a neighborhood of $(0,0)$ hence no computation is needed to see that $f$ is $C^\infty$ at $(0,0)$. More interesting would be the case of $(1-\cos x)^2+y^4$ in the denominator. – Did Jul 12 '15 at 17:34
  • I didnt understand what you meant. – Yinon Eliraz Jul 12 '15 at 17:55
  • @YinonEliraz How is that? – Did Jul 12 '15 at 23:11

1 Answers1

1

HINT:

$\frac{\sin(x)^4 \ln(1+x^2)}{\sqrt{x^2+y^2}} \leq \frac{\sin(x)^4 \ln(1+x^2)}{|x|} = \sin(x)^3 \ln(1+x^2)* \frac {\sin(x)}{|x|}$

${(1+\cos(x))^2+y^4}$ is not so intresting because at $(0,0)$ is not $0$

d_e
  • 1,565