What is the function rule for f(x,y) in terms of s and t?

[beaf123]

<Moderator's note: Moved from homework forum for the general question what it means. For the saddle point question, please re-post it in the homework section, but show us some effort and where you stuck.>

I have been given that

and that

Then I am asked to show, by contradiction, that the point (0, 0) is a saddle point for f.

First my problem is that I have never seen that w is a function x and y in the first equation and a function of s and t in the second equation. How can I insert the second equation into the first?

I am not sure if I have provided enough information, but scream out if you miss something.

 

[GwtBc]

This is essentially just composing functions. For example if you have $f(x)= x^{2}$ and $g(x) = \cos{x}$ then $f(g(x)) = \cos ^{2}{x}$. Here of course there are two variables involved and the functions are a little bit more complicated.

 

[Ray Vickson]

<Moderator's note: Moved from homework forum for the general question what it means. For the saddle point question, please re-post it in the homework section, but show us some effort and where you stuck.>

I have been given that
View attachment 222164

and that

View attachment 222163

Then I am asked to show, by contradiction, that the point (0, 0) is a saddle point for f.

First my problem is that I have never seen that w is a function x and y in the first equation and a function of s and t in the second equation. How can I insert the second equation into the first?

I am not sure if I have provided enough information, but scream out if you miss something.

Just replace $s$ by $x^2$ and $t$ by $x y^2$ when you write the formula for $w(s,t)$.

There are essentially two ways of doing this problem.
(1) Make the replacements above and then deal directly with the function $f(x,y)$ in all its gory detail.
(2) Use the chain rule:
$$\frac{\partial w}{\partial x} = \frac{\partial w}{\partial s} \frac{\partial s}{\partial x}
+ \frac{\partial w}{\partial t} \frac{\partial t} {\partial x}, $$
etc.

Method (2) gets increasingly messy when we go to second derivatives, so if I were doing it I would use Method (1).

 

[PeroK]

First my problem is that I have never seen that w is a function x and y in the first equation and a function of s and t in the second equation. How can I insert the second equation into the first?

I am not sure if I have provided enough information, but scream out if you miss something.

$w$ is a function. Full stop. $x, y, s, t$ are all dummy variables. You could replace them with any other symbols you like.

 

[Ray Vickson]

<Moderator's note: Moved from homework forum for the general question what it means. For the saddle point question, please re-post it in the homework section, but show us some effort and where you stuck.>

I have been given that
View attachment 222164

and that

View attachment 222163

Then I am asked to show, by contradiction, that the point (0, 0) is a saddle point for f.

First my problem is that I have never seen that w is a function x and y in the first equation and a function of s and t in the second equation. How can I insert the second equation into the first?

I am not sure if I have provided enough information, but scream out if you miss something.

You write $w(s,t) = t \cdot (e^{rs} - rs - 1)$.

Is that accurate, or does it contain a "typo"? That is, do you just have some other parameter $r$ involved in the definition of $w$---exactly as you wrote it---or should the $r$ on the right really be $t$?

 

[beaf123]

Thank you fro your answers. I tried to do it with (1) Make the replacements above and then deal directly with the function f(x,y) in all its gory detail. And hopefully I did it right.

It is provided that r>0, so it is just a constant.