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Is there a way to show that if there is a number $c$ for which $g'(c)=0$, then every point on the level set $\{(x,y) \mid h(x,y)=c\}$ is a degenerate critical point of $f$?

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Presumably $f$, $g$, and $h$ have some sort of relationship to each other...? –  Zev Chonoles Mar 28 '12 at 21:40
Well, yes. They are related. –  shmiggens Mar 28 '12 at 22:14
If you don't tell us what the relationship is, it's impossible to answer the question. –  Zev Chonoles Mar 28 '12 at 22:15
I myself do not know. This is a challenge problem given to me by my professor as something to think about. –  shmiggens Mar 28 '12 at 22:22
Let $g(c) = 2c$ or $h(x,y) = c+1$ or $f(x,y) = 0$... Seriously, more conditions are needed. –  Rahul Mar 28 '12 at 22:30

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