True or false questions about partial derivatives

[zhuyilun]

Homework Statement

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 f_xx (2,1)* f_yy (2,1) < (f_xy (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. D_k f(x,y,z)= f_z (x,y,z)

Homework Equations

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?

 

[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

 

[zhuyilun]

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

i) but doesn't f_x and f_y have to be continuous at (a,b)?
iv) can you explain a little bit more about it, i still don't get it

thank you

 

[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

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