Question about Taylor series and implicit function theorem.

furi0n

Hi everyone,
i'll have final exam next week, and have a problem on sheet given by prof.
i attached question as a picture...
respectively,
the first question of sheet is about taylor series as you can see, although we haven't learn about taylor series on two variables, maybe he asked for us to search about this question. do you have any idea (probably have :D ), may you give me hint or same way to solve first question...

In the fifth question, i used implicit functn thrm, i define F(x,y,u,v)=T(x,y)-(u,v)=0, and i found d(u,v)/d(x,y) and d(x,y)/d(u,v). find answer is zero but it should have been zero where do i mistake, is there other choice different from implicit fnctn thrm, then i can understand my mstake or may you help to find my mistake...


in second picture, last question, how can i show the function f has a local min at (0,0) on the line through (0,0). i am considered y=mx line equation as a side condition but this shows me only that (0,0) is a critical point. It's the point i am confused.

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