True or false questions about partial derivatives

1. Oct 15, 2009

zhuyilun

1. The problem statement, all variables and given/known data
1.if the derivative of f(x,y) with respect to x and y both exist, then f is differentiable at (a,b)
2. if (2,1) is a critical point of f and fxx (2,1)* fyy (2,1) < (fxy (2,1))^2, then f has a saddle point at (1,2)
3. if f(x,y) has two local maxima, then f must have a local minimun
4. Dk f(x,y,z)= fz (x,y,z)

2. Relevant equations

3. The attempt at a solution
1. i think it is right, but i can't come up with a good explanation
2. i think it is right according to the second derivative test, but for some reason, the wording of this question keeps making me think this question is wrong
3. i thnk it is right, because if f( x,y) has 2 local max, then there must be a local min between those two local max because the graph must decrease after the first local max
4. i have no idea, can someone explain?

can someone tell me what i did is right or nor?

2. Oct 15, 2009

lanedance

so i assume you just have to decide true or false?

i) what is the defintion of differentiable?
ii) check against 2nd derivative test as you said, rearrange expression to help if necessary, I can't see anything wrong with the wording
iii) that might be true for teh 1 variable case, though imagine two peaks in an otherwise flat plane, is tehre any local minima?
iv) I think this means the directional derivatine in the k (z) direction

3. Oct 15, 2009

zhuyilun

i) but doesn't fx and fy have to be continous at (a,b)?
iv) can you explain a little bit more about it, i still dont get it

thank you

4. Oct 15, 2009

lanedance

i) you still haven't said what the definition of differentiability is?
iv) say you have a vector v, the directional derivative in the direction of a unit vector v is the rate of change of the function moving in the direction of v, it is given by

$$D_{\textbf{v}} = \nabla f \bullet \textbf{v}$$