11.5. Solve the equation $2021 x^{2021}-2021+x=\sqrt[2021]{2022-2021 x}$. (7 points)

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# Solution

The function $f(x)=2021x^{2021}-2021+x$ is increasing, while the function $g(x)=\sqrt[2021]{2022-2021x}$ is decreasing. Therefore, the equation $f(x)=g(x)$ has no more than one root. However, it is obvious that $f(1)=g(1)$.

Answer: $x=1$.

| criteria | points |
| :--- | :---: |
| correct solution | 7 |
| Proven that there is no more than one root. | 4 |
| Root found, but not proven that there are no other roots | 2 |