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一、降低坡度例1 已知:a+b>2,求证:a~4+b~4>2.(英国初中数学竞赛试题)析解:已知条件 a+b 是一次式,待证结论中的 a~4+b~4是四次式,从一次式直接推断四次式,似乎坡度太大,若降低坡度,由二次式入手,则较容易.由 a+b>2,得(a+b)~2>4.①又 (a-b)~2≥0,②①+②得2(a~2+b~2)>4,∴a~2+b~2>2.于是(a~2)~2+(b~2)~2>2,即 a~4+b~4>2.二、有借有还
First, to reduce the gradient Example 1 is known: a + b> 2, verification: a ~ 4 + b ~ 4> 2. (British junior high school math contest questions) analysis: Known condition a + b is a formula, to be confirmed Conclusion The a~4+b~4 is a quadratic type. It is directly inferred from a single formula to a quadratic formula. It seems that the gradient is too large. If the gradient is reduced, it is easier to use a quadratic formula. By a+b>2, (a+b)~2>4.1(ab)~2≥0, 21+2 gets 2(a~2+b~2)>4, ∴a~2+b~2> 2. a~2)~2+(b~2)~2>2, that is, a~4+b~4>2. II.