Example 2. Given that $a, b$ are two unequal positive numbers, among the following three algebraic expressions:
甲. $\left(a+\frac{1}{a}\right)\left(b+\frac{1}{b}\right)$,
乙. $\left(\sqrt{a b}+\frac{1}{\sqrt{a b}}\right)^{2}$,
丙. $\left(\frac{a+b}{2}+\frac{2}{a+b}\right)^{2}$,

the one with the largest value is
(A) Expression 甲;
(B) Expression 乙;
(C) Expression 丙;
(D) It depends on the values of $a, b$.
If we take $a=1, b=2$, then the value of expression 甲 is the largest. If we then take $a=1, b=\frac{1}{3}$, the value of expression 甲 is still the largest. If we conclude that the answer is (A) based on this, it would be incorrect. In fact, if we take $a=3, b=2$, then the value of expression 丙 is the largest, so the correct choice should be (D).