Prove inequality and when does equality hold? Prove that for any real positive numbers $a$ and $b$
$$\left(a+\frac{1}{b}\right)\left(b+\frac{1}{a}\right) \geq 4.$$
When does equality hold?
Hint:

$$ \;\; \text{For any two real numbers $a$ and $b$, } \;\;(ab - 1)^2 \ge 0 $$

Simplify the above expression and try to use the positivity of $a$ and $b$ to arrive at your inequality. And if you do get your inequality then the answer to your follow-up question is pretty much embedded in the hint.
Let's just multiply it out.

$ab + \frac{1}{ab} \ge 2$

Intuïtively we can see that the left expression has minimal value when a=b=1. Its value then is 4.