Question

x+2y=1,求2^x+4^y的最小值

Answer 1

z=2^x+4^y
=2^(1-2y)+4^y
=2*2^(-2y)+4^y
=2*4^(-y)+4^y
=2w+1/w ( 令:w=4^(-y) )
=[ (2w)^(1/2) - (1/w)^(1/2) ]^2 +2*(2w)^(1/2)*(1/w)^(1/2)
=[ (2w)^(1/2) - (1/w)^(1/2) ]^2 +2 * 2^(1/2)
当 (2w)^(1/2) = (1/w)^(1/2) 时，
即 w = 2^(1/2) / 2 = 2^(-1/2)
z有最小值 2* 2^(1/2)=2^(3/2) 即( 2√2 )

又 4^(-y) = w =2^(1/2) / 2
4^(-y) = 2^(-1/2) = 4^(-1/4)
y = 1/4
x = 1-2y =1/2
z(min) = 2^(3/2)

Answer 2

2^x+4^y=2^x+2^2y>=2*(2^x*2^2y)的根号=
2*根号下2^（x+2y）=2根号2
最小值为2根号2

Answer 3

2^x+4^y
=2^x+2^2y>=2√(2^x*2^2y)
=2√2^(x+2y)
=2√2^1
=2√2
所以最小值=2√2

Answer 4

设t=2^x+4^y
t>=2*2^x*4^y
lnt>=ln(2*2^x*4^y)
lnt>=ln2+xln2+2yln2
lnt>=(1+x+2y)ln2
lnt>=ln(2^2)
t>=4