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.

Most answers use functions or derivatives... I'll use another approach: Inequalities!
It's easy to prove that the minimum will be achieved for positive values of $x$ and $y$. Thus, in virtue of the QM-AM inequality: 

$$\sqrt{\frac{x^2+y^2}{2}}\geq \frac{x+y}{2}=5\iff x^2+y^2\geq 50$$


Proof of the Quadratic Mean - Arithmetic Mean inequality ($x,y\geq0)$: $$(x-y)^2\geq 0\iff x^2+y^2\geq 2xy\iff 2(x^2+y^2)\ge (x+y)^2\iff\color{red}{\sqrt{\frac{x^2+y^2}{2}}\geq \frac{x+y}{2}}$$
