求函数f(x)=-()2x+4()x+5的单调递减区间.

求函数f(x)=-()2x+4()x+5的单调递减区间.

答案

思路分析:函数f(x)是复合函数,利用复合函数的单调性求单调递减区间.

解:

定义域是R.

令y=-u2+4u+5,u=()x,

函数y=-u2+4u+5的单调递增区间是(-∞,2],单调递减区间是(2,+∞).

∵u=()x是减函数,

∴函数y=-u2+4u+5是增函数时,f(x)为减函数.

∴u=()x=2-x≤2,得x≥-1.

∴f(x)的单调递减区间为[-1,+∞).

绿色通道:

一般地,对于函数y=af(x),当a>1时,其单调区间和f(x)的单调区间是一致的,并且在相同区间里其增减性是一致的;当0<a<1时,其单调区间和f(x)的单调区间一致,但在相同的区间里其增减性是相反的.

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