HMMT 二月 2007 · CALC 赛 · 第 3 题

英文解析

3. [ 4 ] Let a be a positive real number. Find the value of a such that the definite integral
   ∫2
   d xa
   √
   x + xaachieves its smallest possible value.
   √
   Answer: 3 − 2 2 . Let F ( a ) denote the given definite integral. Then
   ∫2
   d d x 1 1 a
   ′
   F ( a ) = √ = 2 a · √ − √ .
   2 2
   d a x + x a + aa + a
   √a
   √ √ √
   ′ 2
   Setting F ( a ) = 0, we find that 2 a + 2 a = a + 1 or ( a + 1) = 2. We find a = ± 2 − 1, and
   √ √
   √
   because a > 0, a = ( 2 − 1) = 3 − 2 2.2
   x 2