8. Find the minimum value of the function

$$
f(x)=4^{x}+4^{-x}-2^{x+1}-2^{1-x}+5
$$

Solution: Since

$$
\begin{aligned}
4^{x}+4^{-x}-2^{x+1}-2^{1-x}+5=2^{2 x}+ & 2^{-2 x}-2\left(2^{x}+2^{-x}\right)+5= \\
=2^{2 x}+2 \cdot 2^{x} \cdot 2^{-x}+2^{-2 x} & -2-2\left(2^{x}+2^{-x}\right)+5= \\
& =\left(2^{x}+2^{-x}\right)^{2}-2\left(2^{x}+2^{-x}\right)+3
\end{aligned}
$$

then the substitution $t=2^{x}+2^{-x}$ reduces the problem to finding the minimum value of the quadratic polynomial $g(t)=t^{2}-2 t+3$, which is $g(1)=2$.

Answer: 2.
1.I.8. Comment. The following error was made in the given solution. Since $2^{x}>0$, the range of the function $y=2^{x}+2^{-x}$ is the interval $[2 ;+\infty)$. Therefore, it is necessary to find the minimum value of the function $g(t)=t^{2}-2 t+3$ on the interval $[2 ;+\infty)$. On this interval, the function $g(t)$ is increasing, so its minimum value is achieved at $t=2$.

Answer: 3.