1. 函數f(x) = (x^2 + 2) / √(x^2

(x^2 + 2) / √(x^2 + 1)= [(x^2 + 1) + 1] / √(x^2 + 1)= (x^2 + 1) /√(x^2 + 1) + 1/√(x^2 + 1)= √(x^2 + 1) + 1/√(x^2 + 1)= (1/4)√(x^2 + 1) + 1/√(x^2 + 1) + (3/4)√(x^2 + 1) ≥ 2 √ [(1/4)√(x^2 + 1) * 1/√(x^2 + 1)] + (3/4)√(x^2 + 1)= 1 + (3/4)√(x^2 + 1)當且僅當 (1/4)√(x^2 + 1) = 1 / √(x^2 + 1) (x^2 + 1) = 4x = √3 時 , 函數f(x) = (x^2 + 2) / √(x^2 + 1)之最_小_值是 1 + (3/4)√(3 + 1) = 5/2 , 此時x =__√3___.

2010-09-23 23:41:47 補充：
修正 :

(x^2 + 2) / √(x^2 + 1)

= [(x^2 + 1) + 1] / √(x^2 + 1)

= (x^2 + 1) /√(x^2 + 1) + 1/√(x^2 + 1)

= √(x^2 + 1) + 1/√(x^2 + 1)

≥ 2 √ [√(x^2 + 1) / √(x^2 + 1)]

= 2

當且僅當 √(x^2 + 1) = 1/√(x^2 + 1)


x^2 + 1 = 1

x = 0 時 ,

函數f(x) = (x^2 + 2) / √(x^2 + 1)之最_小_值是 2 ,
此時x =__0___.

good boys on this!