①1/(2+√2)+1/(3√2+2√3)+1/(4√3+3√4)+...+1/(2007√2006+2006√2007)②求y=1-2x+2√(4x-3)的最大值

Question

①1/(2+√2)+1/(3√2+2√3)+1/(4√3+3√4)+...+1/(2007√2006+2006√2007)
②求y=1-2x+2√(4x-3)的最大值

Answer 1

① 设有a、b两数,a-b=1,则有
1/(a√b+b√a) = (a√b-b√a) /(a^2 b + b^2 a)
= (a√b-b√a)/1/(ab(a-b)）
= 1/√b - 1/√a
所以
1/(2+√2)+1/(3√2+2√3)+1/(4√3+3√4)+...+1/(2007√2006+2006√2007)
= 1/√1- 1/√2 + 1/√2- 1/√3 +...+ 1/√2006- 1/√2007
= 1 - 1/√2007
② 求y=1-2x+2√(4x-3)的最大值
y = 1-2x+2√(4x-3)
= -1/2 （√(4x-3） -2）^2 + 3/2
最大值是 3/2