For these questions, you should assume the set up for Kantorovich's theorem. That is: f: SR...
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For these questions, you should assume the set up for Kantorovich's theorem. That is: f: SR" is C², S is open, 0 € S, Df (0) is invertible, h=-Df(0)-¹f(0), B(h; h) C S. For question 2 and 3, you should assume f(0)|-L-|Df (0)-¹|² <for some Lipschitz constant L for Df(x). Show that if f: B(h; h)→ R" is a C² function, then Df, is Lipschitz, with Lipschitz constant L₁= max {\Hf,x}, where H denotes the Hessian, and the max is taken over all x € B(h; h). For these questions, you should assume the set up for Kantorovich's theorem. That is: f: SR" is C², S is open, 0 € S, Df (0) is invertible, h=-Df(0)-¹f(0), B(h; h) C S. For question 2 and 3, you should assume f(0)|-L-|Df (0)-¹|² <for some Lipschitz constant L for Df(x). Show that if f: B(h; h)→ R" is a C² function, then Df, is Lipschitz, with Lipschitz constant L₁= max {\Hf,x}, where H denotes the Hessian, and the max is taken over all x € B(h; h).
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