Please help me solve it.

Problem 1: (30 points. Continuation of a discussion from class. It attempts to illustrate the point we raised in class about the difficulty of filling the gap between necessary and sufficient conditions). Let f(z) : R R be an infinitely differentiable function. Let k(x) be the smallest integer that satisfies f)(z) =0, 1 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(z) = oo (i.e. all derivatives vanish at x), butx is a strict isolated local minimum