MT135条件线性化

Posted 2020-10-31 mathstudy

已知\\(x,y>0,\\dfrac{1}{x}+\\dfrac{2}{y}=1\\),求\\(\\dfrac{1}{x+1}+\\dfrac{2}{y+1}\\)的最大值_____

解答：令\\(a=\\dfrac{1}{x},b=\\dfrac{2}{y}\\)则\\(a,b>0,a+b=1\\)
\\(\\dfrac{1}{x+1}+\\dfrac{2}{y+1}=\\dfrac{a}{a+1}+\\dfrac{2b}{b+2}=3-(\\dfrac{1}{a+1}+\\dfrac{4}{b+2})\\le 3-\\dfrac{9}{a+b+3}=\\dfrac{3}{4}\\)

评论:做一个代换，把条件变简单，通过化简，要求的部分变成了我们熟悉的题型.