求函數(shù)y=(1/4)^x -(1/2)^x+1 ,x∈[-3,2]的單調(diào)區(qū)間,并求出它的值域.

數(shù)學(xué)人氣：949 ℃時間：2019-12-13 20:33:58

優(yōu)質(zhì)解答

設(shè) t=(1/2)^x
則y=t^2 - t + 1 = (t-1/2)^2 + 3/4
x∈[-3,2] => t∈[1/4,8]且t=(1/2)^x是減函數(shù)
當(dāng)t=1/2時,y最小值=3/4
當(dāng)t=8時,y最小值=57
故值域y∈[3/4,57]
當(dāng)t∈[1/4,1/2]時y= (t-1/2)^2 + 3/4是減函數(shù)
因t=(1/2)^x也是減函數(shù)
所以函數(shù)單調(diào)增區(qū)間是x∈[1,2]
當(dāng)t∈[1/2,8]時,y= (t-1/2)^2 + 3/4是增函數(shù)
因t=(1/2)^X是減函數(shù)
所以函數(shù)單調(diào)減區(qū)間是x∈[-3,1]