Let [f:] a,b [→E be and open curve of class C∞. For some t∈]a,b [,assume that f′(t)=0, but also that the re exist some integer sp,q with 1≤pq such that f(p)(t) is the first derivative note qualt o 0...

Let [f:] a,b [→E be and open curve of class C∞. For some t∈]a,b [,assume that f′(t)=0, but also that the re exist some integer sp,q with 1≤pq such that f(p)(t) is the first derivative note qualt o 0 and f(q)(t) is the first derivative note qualt o0 and not collinear t of (p) (t). Show that by Taylor’s formula, for h>0 small enough, we have

Where lim h→0,h=0ε (h)=0. As a con sequence, the curve is tangent to the line of direction f(p)(t) passing through f(t). Show that the curve has the following appearance local y at t: