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). 2. Define a₁ = h. Prove that Df (a₁) is invertible, and Df(a₁)-¹| ≤ 2|Df (0)-¹]. Hint: Consider Df(0)-¹ (Df(0) - Df (a₁)). 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). 2. Define a₁ = h. Prove that Df (a₁) is invertible, and Df(a₁)-¹| ≤ 2|Df (0)-¹]. Hint: Consider Df(0)-¹ (Df(0) - Df (a₁)).

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