Max value of sum of squares

Question

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

Answer 1

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.

Answer 2

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:

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 $x = 5$.