maths doubt

[jack]

f(Z)=(4^x)+(2^x)+(4^-x)+(2^-x)
find range?

[Granpa Simpson]

[7,inf)..use AM>=GM..u missed a three I guess..there was a +3 in the question

[jack]

how can we do with AM & GM
Please explain???

[Granpa Simpson]

(4^x+2^x+4^-x+2^-x)/4>=(4^x*2^x*4^-x*2^-x)^(1/4), Now 4^-x=(1/4^x), (2^-x)=(1/2^x), so (4^x*2^x*4^-x*2^-x)=1.
Now, (4^x+2^x+4^-x+2^-x)/4>=1,
(4^x+2^x+4^-x+2^-x)>=4,
(4^x+2^x+4^-x+2^-x)
(4^x+2^x+4^-x+2^-x)>=7,
Hence range=[7,inf)

[jack]

thanxx