Let f be differentiable on an interval A containing zero, and assume (z,,) is a sequence in A with (z,) 0 with z,, # 0. (a) If f(z,) =0 for all n N, show f(0) =0 and f'(0) = 0. (b) Add the assumption that f is twice-differentiable at zero and show that f\"(0) = 0 as well. . Show that the function z 4 22sin(L), ifz#0 g(z) =12 =) : 7 0, ifz=1 is differentiable on R and satisfies '(0) > 0. Prove that is not increasing over any open interval containing 0