切线法切线法是利用切线构造不等式的方法,有时会结合延森不等式。[1]切线法是属于试探性的方法,但使用范围比延森不等式更广,例如半凹半凸的函数 x 2 + 2 x {\displaystyle x^{2}+2{\sqrt {x}}} 不能使用延森不等式,但能使用切线法。[2] 常规方法 对于 x 1 , x 2 , . . . , x n ∈ D , x 1 + x 2 + . . . + x n = k {\displaystyle x_{1},x_{2},...,x_{n}\in D,x_{1}+x_{2}+...+x_{n}=k} ,D为给定区间,k为常数,求证 f ( x 1 ) + f ( x 2 ) + . . . + f ( x n ) ≤ ( ≥ ) C {\displaystyle f(x_{1})+f(x_{2})+...+f(x_{n})\leq (\geq )C} 观察得取等条件为 x 1 = x 2 = . . . = x n = k n {\displaystyle x_{1}=x_{2}=...=x_{n}={\frac {k}{n}}} 时,找出 f ( x ) {\displaystyle f(x)} 在 x = k n {\displaystyle x={\frac {k}{n}}} 处的切线函数 p x + q {\displaystyle px+q} (假设f可导),尝试证明局部不等式 f ( x ) ≤ ( ≥ ) p x + q {\displaystyle f(x)\leq (\geq )px+q} 。[2] 例子 已知 a , b , c ≥ 0 , a + b + c = 1 {\displaystyle a,b,c\geq 0,a+b+c=1} ,求证 4 a + 1 + 4 b + 1 + 4 c + 1 ≤ 21 {\displaystyle {\sqrt {4a+1}}+{\sqrt {4b+1}}+{\sqrt {4c+1}}\leq {\sqrt {21}}} 猜得 a = b = c = 1 3 {\displaystyle a=b=c={\frac {1}{3}}} 时取等,构造切线使 4 a + 1 ≤ p a + q {\displaystyle {\sqrt {4a+1}}\leq pa+q} 让 4 a + 1 + 4 b + 1 + 4 c + 1 ≤ 21 {\displaystyle {\sqrt {4a+1}}+{\sqrt {4b+1}}+{\sqrt {4c+1}}\leq {\sqrt {21}}} 成立。 4 a + 1 = 7 3 {\displaystyle {\sqrt {4a+1}}={\sqrt {\frac {7}{3}}}} ( 4 a + 1 ) ′ = 2 4 a + 1 = 2 3 7 {\displaystyle ({\sqrt {4a+1}})'={\frac {2}{\sqrt {4a+1}}}=2{\sqrt {\frac {3}{7}}}} 所求切线为 2 3 7 ( a − 1 3 ) + 7 3 {\displaystyle 2{\sqrt {\frac {3}{7}}}(a-{\frac {1}{3}})+{\sqrt {\frac {7}{3}}}} ( 2 3 7 ( a − 1 3 ) + 7 3 ) 2 − ( 4 a + 1 ) = 4 21 ( 3 a − 1 ) 2 ≥ 0 {\displaystyle (2{\sqrt {\frac {3}{7}}}(a-{\frac {1}{3}})+{\sqrt {\frac {7}{3}}})^{2}-(4a+1)={\frac {4}{21}}(3a-1)^{2}\geq 0} 证得 4 a + 1 ≤ 2 3 7 ( a − 1 3 ) + 7 3 {\displaystyle {\sqrt {4a+1}}\leq 2{\sqrt {\frac {3}{7}}}(a-{\frac {1}{3}})+{\sqrt {\frac {7}{3}}}} 4 a + 1 + 4 b + 1 + 4 c + 1 ≤ 2 3 7 ( a + b + c − 1 ) + 3 7 3 = 21 {\displaystyle {\sqrt {4a+1}}+{\sqrt {4b+1}}+{\sqrt {4c+1}}\leq 2{\sqrt {\frac {3}{7}}}(a+b+c-1)+3{\sqrt {\frac {7}{3}}}={\sqrt {21}}} [2]参考资料 ^ 程汉波. 对“构造切线法”证明对称不等式的一点改进. 数学教学. 2013, (9) [2014-07-16]. (原始内容存档于2019-05-13). ^ 2.0 2.1 2.2 郭子伟. 例谈不等式证明中的“切线法”及其拓展. 数学空间. 2011, (5): 第27–34页 [2014-07-16]. (原始内容存档于2016-03-04).