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# Symmetric Inequality Problem

I am struggling with understanding https://nrich.maths.org/251.

To restate the problem,

If $$x$$, $$y$$ and $$z$$ are real numbers such that: $$\begin{cases}x+y+z=5 \\ xy+yz+zx=3\end{cases}$$ , What is the largest value that any one of these numbers can have?

In particular, I do not understand the first solution given, and while the second I am getting a grip with (creates a quadratic uses the discriminant inequality since $$x$$, $$y$$ and $$z$$ are real numbers), would like to ask whether any classical inequalities can be used here, as I would be personally more satisfied with this.

The problem I had with applying inequalities I knew was that $$x$$, $$y$$ and $$z$$ could be any real numbers, not just positive.

Any help/discussion would be much appreciated!

Note by Arthur Conmy
1 month ago

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i m getting $$\frac{13}{3}$$ as maximum value and -1 as minimum value · 1 month ago

Yep same here! What was your method? · 1 month ago

but u said u are getting $$\frac{13}{2}$$ . I used the same method which i referred to in the link i gave. · 1 month ago

Oops! Sorry I am getting $$\frac{13}{3}$$. I actually did not check the link out. Let me check it now · 1 month ago

Cauchy Schwarz inequality · 1 month ago

same method as mine then? or in a different way? · 1 month ago

Hey buddy is the answer $$\frac{13}{2}$$? · 1 month ago

okay ! great · 1 month ago

actually u have put a dot by mistake in front of 251 · 1 month ago

The link u have referred is not opening, it says the page not found.However, u can see my solution to this problem : https://brilliant.org/problems/almost-vietas/#!/solution-comments/171217/ · 1 month ago

Thanks Vilakshan. I leant from both your and Sharky's answers. · 1 month ago

Fixed btw · 1 month ago