<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] 
To minimize $f(x,y) = x^2+y^2$; given that $y=10-x$.

Then:
$$f(x,10-x)=x^2+(x-10)^2$$
$f$ is minimized for $f'=0$, so:
$$f'=2x+2(x-10)=0$$
Gives:
$$x=5\land y=5$$
Thus we have:
$$\min (x^2+y^2)=5^2+5^2=50$$
</s>