1. Determine all real solutions of the following equations:
a) $\sqrt[3]{x+1}+\sqrt{x+1}=2$;
b) $\frac{x^{2}+1}{x}=2^{x(2-x)}$.
1. a) It is observed that $x=0$ is a solution and based on the monotonicity of the radical function, it is demonstrated that it is unique.

$3 p$

b) The hypothesis leads to the conclusion $x>0$.

$1 p$

We have $\frac{x^{2}+1}{x} \geq 2$ and $2^{x(2-x)} \leq 2$ for any $x>0$, from which $x=1$.

$3 p$