若x在π/4~π/2,则函数y=
y=tan2x(tanx)^3
=2tanx/[1-(tanx)^2]*(tanx)^2
=2(tanx)^4/[1-(tanx)^2]
=-2[(tanx)^4-(tanx)^2+(tanx)^2-1+1]/[(tanx)^2-1]
=-2{(tanx)^2+1+1/[(tanx)^2-1]}
=-2{[(tanx)^2-1]+1/[(tanx)^2-1]+2}
因为pi/41--->(tanx)^2>1--->(tanx)^2-1>0
依均值不等式[(tanx)^2-1]+1/[(tanx)^2-1]>=2
--->[(tanx)^2-1]+1/[(tanx)^2-1]+2>=4
--...全部
y=tan2x(tanx)^3
=2tanx/[1-(tanx)^2]*(tanx)^2
=2(tanx)^4/[1-(tanx)^2]
=-2[(tanx)^4-(tanx)^2+(tanx)^2-1+1]/[(tanx)^2-1]
=-2{(tanx)^2+1+1/[(tanx)^2-1]}
=-2{[(tanx)^2-1]+1/[(tanx)^2-1]+2}
因为pi/41--->(tanx)^2>1--->(tanx)^2-1>0
依均值不等式[(tanx)^2-1]+1/[(tanx)^2-1]>=2
--->[(tanx)^2-1]+1/[(tanx)^2-1]+2>=4
--->-2{[(tanx)^2=1]+1/[(tanx)^2-1]+2}=[(tanx)^2-1]=1--->(tanx)^2=2
pi/4tanx=√2
所以当x=arctan√2时y=tan2x(tanx)^3有最大值-8。
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