Strict local minimizer (multivariate)

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Discussion Overview

The discussion revolves around the concept of a strict local minimizer in the context of a multivariate function f(x1, x2). Participants are trying to clarify the meaning of the term "minimizer" and whether the statement about having a strict local minimizer at t=0 along every line through (0,0) is valid. The scope includes conceptual clarification and mathematical reasoning.

Discussion Character

  • Conceptual clarification, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant expresses confusion about the term "minimizer," initially interpreting it as "minimum," which leads to a claim that the theorem may not hold true for certain functions, such as f(x,y) = (x-1)² + (y-1)².
  • Another participant attempts to clarify that the minimizer refers to the point t where the minimum occurs, suggesting that the question implies a minimum at t=0 for every function f(x1, x2) along lines defined by x1=at and x2=bt.
  • A participant challenges the clarity of the wording, stating that the phrase "For every f(x1,x2) given that x1 and x2 are lines" is nonsensical, questioning the assumptions about the function f(x1, x2) and its conditions.
  • One participant agrees that the wording of the question is poor and equates a strict local minimizer with a strict local minimum, indicating a need for further clarification from the professor.

Areas of Agreement / Disagreement

Participants express disagreement regarding the interpretation of the term "minimizer" and the validity of the statement about strict local minimizers. There is no consensus on the conditions required for the function f(x1, x2) or the correctness of the original question.

Contextual Notes

There are unresolved assumptions regarding the function f(x1, x2) and its properties, as well as ambiguity in the terminology used in the question.

cateater2000
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I'm stuck on this question

Show that f(x1,x2) has a strict local minimizer at t=0 along every line

{ x1=at
{ x2=bt


through (0,0).


Any hints or tips would be great thanks
 
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What exactly do you mean by "minimizer"? I'm not familiar with that term.

At first I thought you meant "minimum" but in that case the theorem is not true.

Suppose f(x,y)= (x-1)2+ (y-1)2.

Saying that f has a strict local minimum at t= 0 on x= at, y= bt would simply mean that f has a strict local minimum at (0,0) but that is not true- the only minimum of f is at (1,1).
 
The minimizer is the point t where the minimum is. THat's why I'm a bit confused with the question. The wording I was given in my book is a bit awkward.

I think what it means is. For every f(x1,x2) given that x1 and x2 are lines.

There is a minimum at t=0

Those that make sense?
 
cateater2000 said:
The minimizer is the point t where the minimum is. THat's why I'm a bit confused with the question. The wording I was given in my book is a bit awkward.

I think what it means is. For every f(x1,x2) given that x1 and x2 are lines.

There is a minimum at t=0

Those that make sense?

No! In particular, the sentence "For every f(x1,x2) given that x1 and x2 are lines." makes no sense at all. x1, x2 are variables that I presume are numbers, not lines.

Aren't there some conditions on the function f(x1, x2)? The statement is certainly NOT true for general f.
 
I think you're right. The wording of the question is not very good. A strict local minimizer is in fact the same as a strict local minimum.

I geuss I'll have to ask the prof
 

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