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

  • $\begingroup$ 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

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.

  • $\begingroup$ 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

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 $x = 5$.

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.