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Math Help - Is this proof adequate? [max f(x) = -min(-f(x))]. Also 2question about Newton Method

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    Is this proof adequate? [max f(x) = -min(-f(x))]. Also 2question about Newton Method

    So I'm supposed to prove that max f(x) = -min(-f(x)) and I wonder if my proof is okay mathematically.



    We see that the same x, let's call it xmin, maximizes f(x) that minimizes -f(x) =>

    max f(x) = f(xmin)
    max (-f(x)) = -f(xmin)

    =>

    max f(x) = f(xmin)
    -min (-f(x)) = f(xmin)

    => max f(x) = -min(-f(x))

    My concern is the first step when I use the figure to "see" that it is the same x that maximizes f(x) that minimizes -f(x).

    I've also got a question regarding Newton method and Steepest descent method: if Newton method converges, does it always converge faster than SD?

    Also a question regarding Newton method in 2 dimensions: how far away can the initial starting point be from the minimum before we start loosing the convergence to the minimum?

    Thanks! (First post )
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  2. #2
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    Re: Is this proof adequate? [max f(x) = -min(-f(x))]. Also 2question about Newton Met

    your picture is a convincing plausability argument, but it's not mathematically rigourous.
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