Please refer to the picture attached

Let f : R" - R be continuously differentiable over R". Show that f is conver if and only if (Vf(y) -Vf(x),y- x) 20 for allx, y e Rn Hint: It can be shown by proving that the linearization relation in Exercise 1 is equivalent to the gradient monotonicity of this exercise. A relation that may be useful is: for a differentiable scalar function o : [0, 1] -> R, we have $(1) - $(0) = p' (t) dt o and then use d(a) = f(x +a(y - x))