Taylor's formula for two variables

[Nikitin]

http://sphotos-h.ak.fbcdn.net/hphotos-ak-ash4/485580_10200547582988715_854455727_n.jpg

In formula 1 it says F(1) = F(0)+F'(0)+ 0.5F''(C)
Where the heck dos the C come from? I thought they were applying taylor's formula to find an approximation of F(1), around t=0. Then c=0, right? Is it because h and k are very small?

In other words, how is there room for a c, when x=1 and x₀=0?

 

[Nikitin]

http://sphotos-a.ak.fbcdn.net/hphotos-ak-prn1/945518_10200547626669807_1364716261_n.jpg

At point 4, why is another test needed? Since the term (h*f_xx+k*f_xy)² is always positive, then term Q(0) must always have the same sign as f_xx, no?

 

[Nikitin]

I figured out the question in post 1, but I still don't understand the1 in post 2.

 

[HallsofIvy]

http://sphotos-a.ak.fbcdn.net/hphotos-ak-prn1/945518_10200547626669807_1364716261_n.jpg

At point 4, why is another test needed? Since the term (h*f_xx+k*f_xy)² is always positive, then term Q(0) must always have the same sign as f_xx, no?

No, it could be 0, as is pointed out in your insert.

 

[Nikitin]

oh,of course. thanks!