Let f(x) be a real-valued function whose Hessian matrix ( ∂2

∂xi∂xj f)

is positive definite throughout some convex open set U of Rm. For u = 0 and x ∈ U, show that the function t → f(x + tu) of the real variable t is strictly convex on {t : x + tu ∈ U}. Use this fact to demonstrate that f(x) can have at most one local minimum point on any convex subset of U.