Strict local minimizer (multivariate)

1. Oct 19, 2004

cateater2000

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

2. Oct 20, 2004

HallsofIvy

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).

3. Oct 20, 2004

cateater2000

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?

4. Oct 21, 2004

HallsofIvy

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.

5. Oct 21, 2004

cateater2000

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