5. Let f RR be twice differentiable. Prove that the following are equivalent: (a). f is...

Transcribed Image Text:

5. Let f RR be twice differentiable. Prove that the following are equivalent: (a). f is convex, that is, (Ax + (1 A)y) f (x) + (1 )(y), \\ [0, 1], Vx, y R. - (b). For all x, y R, (y) f(x) + f'(x)(y x). Use this to show that (f'(y) f'(x))(y x) 0, and conclude from this that f' is monotonically increasing. 4 (c). f" 0. Hint: At some point you might want to consider the function g(A) = A (x) + (1 ) (y) (x + (1 A)y), - and study g'(x). 5. Let f RR be twice differentiable. Prove that the following are equivalent: (a). f is convex, that is, (Ax + (1 A)y) f (x) + (1 )(y), \\ [0, 1], Vx, y R. - (b). For all x, y R, (y) f(x) + f'(x)(y x). Use this to show that (f'(y) f'(x))(y x) 0, and conclude from this that f' is monotonically increasing. 4 (c). f" 0. Hint: At some point you might want to consider the function g(A) = A (x) + (1 ) (y) (x + (1 A)y), - and study g'(x).

Answer rating: 100% (QA)

To solve the system using the Jacobi method we need to split the system into diagonal and offdiagonal components The diagonal components consist of th... View the full answer