对于任意的x1,x2∈R.若函数f(x)=2^x,试比较[f(x1)+f(x2)]/2与f[(x1+x 2)/2]的大小
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对于任意的x1,x2∈R.若函数f(x)=2^x,试比较[f(x1)+f(x2)]/2与f[(x1+x 2)/2]的大小
![对于任意的x1,x2∈R.若函数f(x)=2^x,试比较[f(x1)+f(x2)]/2与f[(x1+x 2)/2]的大小](/uploads/image/z/15920046-54-6.jpg?t=%E5%AF%B9%E4%BA%8E%E4%BB%BB%E6%84%8F%E7%9A%84x1%2Cx2%E2%88%88R.%E8%8B%A5%E5%87%BD%E6%95%B0f%28x%29%3D2%5Ex%2C%E8%AF%95%E6%AF%94%E8%BE%83%5Bf%28x1%29%2Bf%28x2%29%5D%2F2%E4%B8%8Ef%5B%28x1%2Bx+2%29%2F2%5D%E7%9A%84%E5%A4%A7%E5%B0%8F)
底数2>1,2^x恒>0
[f(x1)+f(x2)]/2/f[(x1+x2)/2]
=[(2^x1+2^x2)/2]/2^[(x1+x2)/2]
=[2^(x1-1)+2^(x2-1)]/2^[(x1+x2)/2]
=2^[x1-1 -(x1+x2)/2] +2^[x2-1-(x1+x2)/2]
=2^[(x1-x2)/2 -1]+2^[(x2-x1)/2 -1]
=(1/2)[2^[(x1-x2)/2] +1/2^[(x1-x2)]/2]
由均值不等式得2^[(x1-x2)/2]+1/2^[(x1-x2)/2]≥2,当且仅当x1=x2时取等号.
(1/2)[2^[(x1-x2)/2] +1/2^[(x1-x2)]/2]≥1
[f(x1)+f(x2)]/2≥f[(x1+x2)/2],当且仅当x1=x2时取等号
[f(x1)+f(x2)]/2/f[(x1+x2)/2]
=[(2^x1+2^x2)/2]/2^[(x1+x2)/2]
=[2^(x1-1)+2^(x2-1)]/2^[(x1+x2)/2]
=2^[x1-1 -(x1+x2)/2] +2^[x2-1-(x1+x2)/2]
=2^[(x1-x2)/2 -1]+2^[(x2-x1)/2 -1]
=(1/2)[2^[(x1-x2)/2] +1/2^[(x1-x2)]/2]
由均值不等式得2^[(x1-x2)/2]+1/2^[(x1-x2)/2]≥2,当且仅当x1=x2时取等号.
(1/2)[2^[(x1-x2)/2] +1/2^[(x1-x2)]/2]≥1
[f(x1)+f(x2)]/2≥f[(x1+x2)/2],当且仅当x1=x2时取等号
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