關於四項時的柯西不等式 x 1 , x 2 , y 1 , y 2 {\displaystyle x_{1},x_{2},y_{1},y_{2}} 都是非0實數,我們有 ( x 1 2 + x 2 2 ) ( y 1 2 + y 2 2 ) ≥ ( x 1 y 1 + x 2 y 2 ) 2 {\displaystyle (x_{1}^{2}+x_{2}^{2})(y_{1}^{2}+y_{2}^{2})\geq (x_{1}y_{1}+x_{2}y_{2})^{2}} ,若且唯若 x 1 y 1 = x 2 y 2 {\displaystyle {\frac {x_{1}}{y_{1}}}={\frac {x_{2}}{y_{2}}}} ,等號成立。 現在令 x 1 = y 1 = a , x 2 = y 2 = b {\displaystyle x_{1}=y_{1}=a,x_{2}=y_{2}=b} ,則 ( a 2 + b 2 ) ( a 2 + b 2 ) ≥ ( a × a + b × b ) 2 {\displaystyle (a^{2}+b^{2})(a^{2}+b^{2})\geq (a\times a+b\times b)^{2}} ,化簡得 ( a 2 + b 2 ) 2 ≥ ( a 2 + b 2 ) 2 {\displaystyle (a^{2}+b^{2})^{2}\geq (a^{2}+b^{2})^{2}} 請問此時不等式仍然是正確的嗎?如何解釋「若且唯若 a a = b b {\displaystyle {\frac {a}{a}}={\frac {b}{b}}} ,等號成立。」裡的「唯若」?難道 a a = b b {\displaystyle {\frac {a}{a}}={\frac {b}{b}}} 有不成立的時候嗎?謝謝解答!