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Question is

Sum of two numbers x and y is 10. What is the maximum and minimum value of its sum of squares.

How can I find the maximum?

The minimum turns out to be 50 using AM GM

Please do try to solve this using AM GM

Please help me out..

Yeah I found its min value and then I went on a long way for it's max value but failed

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  • $\begingroup$ Have you tried anything? $\endgroup$ – Parcly Taxel Oct 21 '18 at 12:14
  • $\begingroup$ Yeah I found its min value and then I went on a long way for it's max value but failed $\endgroup$ – user606630 Oct 21 '18 at 12:15
  • $\begingroup$ @ParclyTaxel I tried using quadratic means and harmonic means. Can you please help me out. I am not able to go on any further. $\endgroup$ – user606630 Oct 21 '18 at 12:17
  • $\begingroup$ I need restrictions on $x,y$. Are they positive? Nonnegative? $\endgroup$ – Parcly Taxel Oct 21 '18 at 12:20
  • $\begingroup$ @ParclyTaxel Oops I am sorry I forgot to mention it. They are non negative $\endgroup$ – user606630 Oct 21 '18 at 12:21
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If we take AM $\ge$ GM for $x^2$ and $y^2$, we get

$\frac {x^2+y^2} 2 \ge \sqrt{x^2y^2}$

$x^2+y^2 \ge 2xy$

$\implies$ $(x+y)^2 - 2xy \ge 2xy$

$\implies$ $100 \ge 4xy$

$\implies$ $xy \le 25$

Here you can find max value of $xy$, you already have found min value of $x^2+y^2$.

Max value of $x^2+y^2$ does NOT exist.

You may understand this as, the graph being parabolic so no global extrema.

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  • $\begingroup$ Good solution but the book is asking for the max value using AM GM inequality. I don't know Co ordinate. $\endgroup$ – user606630 Oct 21 '18 at 14:23
  • $\begingroup$ If you can look into it and if it does not exist then give me a proof other than parabola as I don't know what it is. $\endgroup$ – user606630 Oct 21 '18 at 14:24
  • $\begingroup$ @user606630, parabola is the graph that is formed for any quadratic equation. Quadratic equation is of form $ax^2 + bx + c$ where $a$ is non zero. Read more about Parabola here, google.com/… $\endgroup$ – PradyumanDixit Oct 21 '18 at 15:47
  • $\begingroup$ @user606630 check the question once more, if you find that you are really asked the maximum value of $x^2+y^2$, say infinity or does not exist. You may try the maximising and minimising the function Jose gave in his answer, you will get the same answer. $\endgroup$ – PradyumanDixit Oct 21 '18 at 15:53
  • $\begingroup$ @user606630 did this help you? $\endgroup$ – PradyumanDixit Oct 22 '18 at 12:04
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Hint: $x+y=10$, hence you ought to minimize and maximize the function $$f(x) = x^2 + (10 - x)^2 = 2x^2 - 20x + 100 = 2(x^2 - 10x + 50).$$

Can you take it from here?

Added December 23 2018

A plot of the graph of $f(x) = 2(x^2 - 10x + 50)$ taken from WolframAlpha is as follows: enter image description here

Clearly, the maximum value ($100$) of $f$ occurs at the endpoints $x = 0$ and $x = 10$, while the minimum may be computed via the critical number $c$ as follows: $$f'(x) = 2(2x - 10)$$ $$f'(c) = 0 = 2(2c - 10)$$ $$c = 5$$ $$f''(x) = 4 > 0$$

Therefore, the minimum value ($50$) of $f$ occurs at $x = 5$.

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  • $\begingroup$ No no we only can use AM GM inequality $\endgroup$ – user606630 Oct 21 '18 at 12:24

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