Max value of sum of squares

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

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

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

• Good solution but the book is asking for the max value using AM GM inequality. I don't know Co ordinate. – user606630 Oct 21 '18 at 14:23
• 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. – user606630 Oct 21 '18 at 14:24
• @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/… – PradyumanDixit Oct 21 '18 at 15:47
• @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. – PradyumanDixit Oct 21 '18 at 15:53

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?

A plot of the graph of $$f(x) = 2(x^2 - 10x + 50)$$ taken from WolframAlpha is as follows:
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 $$x = 5$$.