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

Find the range of the function $f(x)=x-1+\sqrt{6 x-x^{2}}$.

Fourth Olympiad (2010). I (correspondence) round
Solution 1. The domain of the given function is the interval $[0 ; 2]$. To find the set of its values, we need to find the points where the derivative of this function is zero. Since $f^{\prime}(x)=1+\frac{3-3 x}{\sqrt{6 x-x^{2}}}$, then

$$
f^{\prime}(x)=0 \Longleftrightarrow \sqrt{6 x-3 x^{2}}=3 x-3 \Longleftrightarrow\left\{\begin{array}{l}
4 x^{2}-8 x+3=0 \\
x \geqslant 1
\end{array}\right.
$$

from which $x=\frac{3}{2}$. We will find the values of this function at $x=\frac{3}{2}$ and at the endpoints of the interval $[0 ; 2]$ - the domain of this function.

| $x$ | 0 | $\frac{3}{2}$ | 2 |
| :---: | :---: | :---: | :---: |
| $f(x)$ | -1 | 2 | 1 |

Therefore, the set of values of the function is the interval $[-1 ; 2]$. Topic: "Application of the derivative for function research"