Let f(x) RR be an infinitely differentiable function. Let k(x) be the smallest integer that satisfies f(j) (x) = 0, 1 j k 1, (k) (x) 0 (that is the smallest index of the derivative that is nonzero). Here I denote by f() (x) the 1-th derivative of f at x. I. Prove that, if k(x) is odd, then x is neither a local minimum nor a local maximum. (For all these steps I suggest you use Taylor's theorem for the appropriate order, you can use the form in wikipedia). II. Prove that if k(x) is even then x is either a strict local minimum or a strict local maximum. (it is sufficient to show it is a minimum if f(k) (x) > 0) III. In the same situation, prove that x is an isolated extremum (hint: use the optimality conditions and apply Taylor's theorem to the derivative of f). IV. Give an example where k(x) = (i.e. all derivatives vanish at x), but x is a strict isolated local minimum.