x & gt0
2^x- 1/2^x=2
2^x= 1+√2
x=log2( 1+√2)
x & lt0
2^x- 1/2^-x=2
2 x-2 x = 2 is invalid.
g(t)=2^tf(2t)+mf(t)
=2^t[2^(2t)- 1/2^(2t)]+m(2^t- 1/2^t)
=2^(3t)- 1/2^t+m(2^t- 1/2^t)
x=2^t
2 & lt= x & lt=4
4 & lt=x^2<; = 16
g(x)=x^3- 1/x+m(x- 1/x)=(x^4+mx^2- 1-m)/x>; =0
x^4+mx^2- 1-m>; =0
x & gt0
x^2>; =-m/2+√(m^2+4+4m)/2=-m/2+|m+2|/2
x^2<; =-m/2-√(m^2+4+4m)/2=-m/2-|m+2|/2
m & lt-2
x^2>; =-m/2+√(m^2+4+4m)/2=-m/2-m/2- 1=-m- 1<; 4 does not hold water
x^2<; =-m/2-√(m^2+4+4m)/2=-m/2+m/2+ 1= 1<; 4 does not hold water
m & gt=-2
x^2>; =-m/2+√ (m 2+4+4m)/2 =-m/2+m/2+1=1hold.
x^2<; =-m/2-√(m^2+4+4m)/2=-m/2-m/2- 1=-m- 1<; = 16
-m- 1 & lt; = 16
m & gt=-2