Question

求函数y=3(4^x+4^-x)-10(2^x+2^-x)的最小值.求完整过程~

Answer 1

y = 3[4^x+4^(-x)] - 10[2^x+2^(-x)]

= 3{[2^x+2^(-x)]^2-2} - 10[2^x+2^(-x)]

= 3[2^x+2^(-x)]^2 - 10[2^x+2^(-x)] - 6

= 3{[2^x+2^(-x)]-5/3}^2 - 43/3

2^x+2^(-x) = {√[2^x]-1/[√(2^x)]}^2+2≥2

[2^x+2^(-x)]-5/3 ≥1/3

3{[2^x+2^(-x)]-5/3}^2 ≥ 1/3

3{[2^x+2^(-x)]-5/3}^2 - 43/3 ≥ -14

y=3(4^x+4^-x)-10(2^x+2^-x)的最小值-14