3. Find all values of $m$ for which any solution of the equation

$$
2019 \cdot \sqrt[3]{3.5 x-2.5}+2018 \cdot \log _{2}(3 x-1)+m=2020
$$

belongs to the interval $[1 ; 3]$.
Answer: $m \in[-8072 ;-2017]$.

## Solution:

Consider $f(x)=2019 \cdot \sqrt[3]{3.5 x-2.5}+2018 \cdot \log _{2}(3 x-1)+m-2020$.

On the domain of definition $D_{f}=\left(\frac{1}{3} ;+\infty\right)$, the function is monotonically increasing, as the sum of monotonically increasing functions. The equation $f(x)=0$ has a unique solution. This solution will belong to the interval $[1 ; 3]$ if and only if,

when $\left\{\begin{array}{l}f(1) \leq 0, \\ f(3) \geq 0 .\end{array} \Rightarrow\left\{\begin{array}{c}2019+2018+m-2020 \leq 0, \\ 2019 \cdot 2+2018 \cdot 3+m-2020 \geq 0 .\end{array} \Rightarrow\left\{\begin{array}{l}m \leq-2017, \\ m \geq-8072 .\end{array}\right.\right.\right.$