Newton-Raphson method with two x values

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The discussion focuses on applying the Newton-Raphson method to minimize the function f = x1^2 + x2^2 + 10x1 + 20x2 + 25. The user successfully splits the problem into two separate functions: f(x1) = x1^2 + 10x1 + 5 and f(x2) = x2^2 + 20x2 + 25. By utilizing the Newton-Raphson method for each function independently, the user confirms that this approach is valid due to the absence of cross-terms involving both x1 and x2 in the original function.

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Find the value of x1 and x2 that minimize the function f= x1^2 + x2^2 + 10x1 +20x2 +25 . This isn't that difficult of a problem if there was only 1 x value. But how do I do it when it is using x1 and x2?
 
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Okay I think I might have figured something out. I split the problem up to be f(x1)=x1^2+10x1+5 and f(x2)=x2^2+20x2+25 and then do Newton-raphson for each of those functions. Is this correct?
 
In this case, it's going to pretty easy since there aren't any terms in the expression for [itex]f(x_1,x_2)[/itex] that involve both [itex]x_1[/itex] and [itex]x_2[/itex] so you can indeed do this one by finding the minimums with respect to [itex]x_1[/itex] and [itex]x_2[/itex].
 

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