<s>[INST] Minimum point of $x^2+y^2$ given that $x+y=10$ How do you I approach the following question:
Find the smallest possible value of $x^2 + y^2$ given that $x + y = 10$.
I can use my common sense and deduce that the minimum value is $5^2 + 5^2 = 50$. But how do you approach this mathematically?
Thanks in advance.
 [/INST] $\large \text{How about this approach:}$
$
x + y = 10, \text{ so } y = 10 - x \\
x^2 + y^2 = k, y = \sqrt{k - x^2}\\
\rightarrow 10 - x = \sqrt{k - x^2}\\
\rightarrow 100 - 20x + x^2 = k - x^2\\
\rightarrow 2x^2 - 20x + 100 = k\\
f\prime = 4x - 20\\
\text{When x is 5, } f\prime = 0\\~\\
\text{Testing values on the right and left: }\\
f(4) = 62, f(6) = 52\\~\\
\therefore \textbf{There is a minimum point at } x = 5, \textbf{ hence } y = 5.\\
5^2 + 5^2 = 50
$
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