Question 78
For $a$ and $b$ real numbers, define
$$
E(a,b) = \sqrt{\left(\frac{1}{2} - a\right)^2 + \left(\frac{1}{2} - b\right)^2} + \sqrt{\left(\frac{1}{2} + a\right)^2 + \left(\frac{1}{2} + b\right)^2}.
$$
a) Prove that $E(a, b) \ge \sqrt{2}$, for every real numbers $a, b$, and equality takes place if and only if $a = b \in [-\frac{1}{2}, \frac{1}{2}]$.
b) Find $\min \{E(a, a+2) \mid a \in \mathbb{R}\}$.
$$
E(a,b) = \sqrt{\left(\frac{1}{2} - a\right)^2 + \left(\frac{1}{2} - b\right)^2} + \sqrt{\left(\frac{1}{2} + a\right)^2 + \left(\frac{1}{2} + b\right)^2}.
$$
a) Prove that $E(a, b) \ge \sqrt{2}$, for every real numbers $a, b$, and equality takes place if and only if $a = b \in [-\frac{1}{2}, \frac{1}{2}]$.
b) Find $\min \{E(a, a+2) \mid a \in \mathbb{R}\}$.