HMMT 十一月 2010 · 冲刺赛 · 第 4 题

英文解析

4. [ 6 ] Let x be a real number. Find the maximum value of 2 .
   √
   y y 4
   Answer: 2 Consider the function 2 . This is monotonically increasing, so to maximize 2 , yousimply want to maximize y . Here, y = x (1 − x ) = − x + x is a parabola opening downwards. The 2
   vertex of the parabola occurs at x = ( − 1) / ( − 2) = 1 / 2, so the maximum value of the function is
   √
   (1 / 2)(1 / 2)4
   2 = 2.