Is the following function differentiable?

1. Dec 14, 2011

gipc

I have:
http://img12.imageshack.us/img12/6121/capturerhf.png [Broken]

Is the function differentiable in (0,2)? If so, find its Tangent Plane.

So far I have
We have $(\nabla f)(0,2)=(f_x(0,2).f_y(0,2))=\ldots=(0,1)$ , so if $f$ is differentiable at $(0,2)$ the only possible differential is $\lambda (h,k)=(\nabla f)(0,2)(h,k)^t=k$ .So I have to analyze $\displaystyle\lim_{(h,k) \to (0,0)} \frac{|f(0+h,2+k)-f(0,2)-\lambda (h,k)|}{ \left\|{(h,k)}\right\|}=0$ but I can't seem to solve it.

And I also don't know how to find a tangent plane

Last edited by a moderator: May 5, 2017
2. Dec 14, 2011

gipc

I came to a conclusion that f is indeed differentiable.

Can someone please help me understand how to prove it? Someone suggested that i use the Taylor expansion but i don't know how to use it. So i'm hoping someone could show me :)

3. Dec 14, 2011

D H

Staff Emeritus
As a prerequisite, the function needs to be continuous at the point in question for a derivative to exist at that point. Is this function continuous at (x,y)=(0,2)?

Assuming it is continuous, this is a function of two variables, so you need to investigate both the partial with respect to x and with respect to y. For x not equal to zero you will get one pair of partial derivatives, for x equal to zero you will get another. Is the first set equal to the second set in the limit x→0?

4. Dec 14, 2011

gipc

when i try to take the partial derivatives i can't really take them to x->0 and i get stuck with the epsilon business.

The general steps for showing differentiability doesn't really apply here.