不等式a~2+b~2≥2ab(1)在数学中有着广泛的应用,但有时直接用这个不等式解题,往往会导致错误甚至思路受阻。如果我们在(1)式的两边同时加上左边式子,则有2(a~2+b~2)≥a~2+b~2+2ab,整理可得a~2+b~2≥((a+b)~2)/2(2),比较(1)式和(2)式的右边,易知(a+b)~2/2≥2ab,所以不等式(2)则是不等式(1)的一个加强形式。例1已知a>0,b>0,且a+b=1,求(a+1/b)~2+(b+1/a)~2的最小值。 Inequality a ~ 2 + b ~ 2 ≥ 2ab (1) has a wide range of applications in mathematics, but sometimes directly solve the problem with this inequality, often lead to errors or even blocked ideas. If we add the left-hand side of both sides of (1), we have 2 (a ~ 2 + b ~ 2) ≥a ~ 2 + b ~ 2 + 2ab, finishing a ~ 2 + b ~ 2≥ (a + b) to 2/2 / 2 (2), the right side of the equations (1) and (2) is compared and it is easy to know that the inequality (1) an enhanced form. Example 1 Knowing a> 0, b> 0, and a + b = 1, find the minimum value of (a + 1 / b) ~ 2 + (b + 1 / a) ~ 2.