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

$$
2018 \cdot \sqrt[5]{6.2 x-5.2}+2019 \cdot \log _{5}(4 x+1)+m=2020
$$

belongs to the interval $[1 ; 6]$.
# Answer: $m \in[-6054 ;-2017]$.

## Solution:

Consider $f(x)=2018 \cdot \sqrt[5]{6.2 x-5.2}+2019 \cdot \log _{5}(4 x+1)+m-2020$.

On the domain of definition $D_{f}=\left(-\frac{1}{4} ;+\infty\right)$, 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 $[1 ; 6]$ if and only if,

$$
\left\{\begin{array} { l } 
{ f ( 1 ) \leq 0 , } \\
{ f ( 6 ) \geq 0 . }
\end{array} \Rightarrow \left\{\begin{array} { c } 
{ 2 0 1 8 + 2 0 1 9 + m - 2 0 2 0 \leq 0 } \\
{ 2 0 1 8 \cdot 2 + 2 0 1 9 \cdot 2 + m - 2 0 2 0 \geq 0 . }
\end{array} \Rightarrow \left\{\begin{array}{l}
m \leq-2017 \\
m \geq-6054
\end{array}\right.\right.\right.
$$