For positive real numbers a and b, the least value of (a + b)(1/a + 1/b) is:

Expanding, $(a+b)(1/a+1/b)=1+a/b+b/a+1=2+(a/b+b/a)$. Since AM ≥ GM applied to the positive numbers a/$b$ and $b$/a gives a/$b+b$/a ≥ 2√1 = 2, the whole expression is at least 4, with equality when $a=b$. (2 is only the constant part, before the AM–GM bound is added.) Test $a=b=1:2\times 2=4$, while $a=1,b=3$ gives $4\times (4/3)=16/3>4$.