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