(a) Let g: (a,b) → R be continuous and let g(x) = c> 0 for a point xo € (a,b). Show that there exists 8 > O such that g(0) > 0 holds for all & € (x0 - 8, X0 + 8) (a,b). (5 marks)
(b) Let f: (a,b) - R be a 4-times continuously differentiable function. Let x0 € (a,b) with f'(x) = f'(x) = f'(X) = 0 and c := = f(x) > 0. [10 marks) Show that f has a local minimum at Xo. Hint: Use Lagrange's version of Taylor's theorem and part (a) for g(x) = f(x). (c) Show that the function f(x) = x2 + 2 cos x has a local minimum at 0. [5 marks) +