Let f and g be smooth real-valued functions of x R n ; for any real number M, let xM be optimal for min g(x)=M f(x); and assume g(xM) = 0. (a) It is a fact from advanced calculus that in the vicinity of xM, {x : g(x) = M} is a smooth (n 1)-dimensional surface whose tangent plane at xM is the orthogonal complement of g(xM); in particular, for every d R n such that g(xM)d = 0 there is a curve (s) defined near s = 0 such that (0) = xM, g((s)) = M for all s, and (0) = d. Using this, show that f(xM)d = 0 whenever d R n satisfies g(xM)d = 0