11.2 Solve the equation $3 \sin x + 4 \cos x = 2^{x+3} + 2^{-x}$.
Answer: No roots. Solution. The left side of the equation, by introducing an auxiliary angle, is reduced to the form $5 \sin (x+\alpha)$. This means it does not exceed 5. For the right side, by the inequality between the arithmetic mean and the geometric mean (for positive numbers), we will have: $2^{x+3}+2^{-x} \geq 2 \sqrt{2^{x+3} \cdot 2^{-x}}=2 \sqrt{8}=4 \sqrt{2}>5$.