Problem 1. How many solutions does the equation $\sqrt{x}+x^{3}=5$ have?
Solution. The function $f(x)=\sqrt{x}+x^{3}$ is defined for $x \geqslant 0$, continuous, monotonic, non-negative, and varies from 0 to $+\infty$. Therefore, the given equation has only one solution $x_{1}$. Obviously, $1<x_{1}<2$.