6. Solve the following problem using as many methods as possible. Methods are considered different if they use different mathematical ideas, as well as various technical approaches to implementing the same idea. Indicate the place of each method used in the school mathematics curriculum.

Determine (depending on the value of the parameter $a$) the number of solutions to the equation $\sqrt{x}+\sqrt{6-2 x}=a$.
Solution 1. The given function is defined and continuous on the interval $[0 ; 3]$. Let's compute its derivative

$$
f^{\prime}(x)=\frac{1}{2 \sqrt{x}}-\frac{1}{\sqrt{6-2 x}}=\frac{\sqrt{6-2 x}-2 \sqrt{x}}{2 \sqrt{x(6-2 x)}}.
$$

By solving the inequality $\sqrt{6-2 x} \geqslant 2 \sqrt{x}$, we get that $x \leqslant 1$. Thus, the function $f(x)=\sqrt{x}+\sqrt{6-2 x}$ is increasing on the interval $[0 ; 1]$ and decreasing on the interval $[1 ; 3]$. The table below shows its values at the endpoints of the interval $[0 ; 3]$ and at the point $x=1$.

| $x$ | 0 | 1 | 3 |
| :---: | :---: | :---: | :---: |
| $f(x)$ | $\sqrt{6}$ | 3 | $\sqrt{3}$ |

The graph of the considered function is shown in the following figure,

![](https://cdn.mathpix.com/cropped/2024_05_21_46613852391e49da1fa9g-038.jpg?height=517&width=557&top_left_y=1626&top_left_x=767)

from which it is clear that the equation has one root when $a=3$ and $a \in[\sqrt{3} ; \sqrt{6})$, and has two roots when $a \in[\sqrt{6} ; 3)$.

Topic: "Application of Derivatives for Function Analysis"

38