Supposed that f: D -> R is a function defined at a point xo so that f'(xo) = 0 or f'(xo) does not exist. Also suppose that f is differentiable on an open interval containing xo except possibly at xo and that f is continuous at xo.
Then prove that if f'(x)>0 on some interval (x1,x0) and f'(x)<0 on some interval (xo,x2) then f has a relative maximum at xo.