Formulate and solve algebraically, through optimal conditions
first order, the problem of finding the point of the curve $x_{2} = x_{1}(3 − x_{1})$ that is closest to the point $(3 3)^{T}$. What is the guarantee that the point obtained is in fact the desired solution? Explain. Suggestion: explore the geometric visualization of the elements of the problem to assist you in algebraic analysis.
First part of the solution:
The proposed problem is equivalent to
minimize $(x_{1} − 3)^{2} + (x_{2} − 3)^{2}$
subject to $x^{2}_{1} − 3x^{1} + x_{2} = 0$.
My question is, how the author got there $f(x) =(x_{1} − 3)^{2} + (x_{2} − 3)^{2}$ ?
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It's just the distance formula. When $\sqrt{f(x)}$ is minimised (write it out in terms of $x_1,x_2$), $f(x)$ is minimised as well.
Toby Mak
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Thank you very much, this problem has a global solution, can you tell me how the author saw this solution right away? – Amissadai ferreira Oct 17 '20 at 03:35
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Through practice. If you are familiar with coordinate geometry, you will see this right away. – Toby Mak Oct 17 '20 at 03:43
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I sketched the circumference and the constraint function (which is a parabola), and saw that the circumference touches a point in the parabola. Would this point be a global solution? – Amissadai ferreira Oct 17 '20 at 03:53
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Yes, this would be a global solution. – Toby Mak Oct 17 '20 at 03:56
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Thank you very much. – Amissadai ferreira Oct 17 '20 at 04:08