Example. Find the maximum and minimum values of the function

$$
y=\frac{10 x+10}{x^{2}+2 x+2}
$$

on the interval $[-1,2]$.
Solution.

1. The given function is continuous on the interval $[-1,2]$, as it is the ratio of continuous functions with a denominator not equal to zero $\left(x^{2}+2 x+2>0\right)$.
2. We calculate the derivative of the given function:

$$
y^{\prime}=-10 \frac{x^{2}+2 x}{\left(x^{2}+2 x+2\right)^{2}}
$$

3. The critical points of the given function are $x=0$ and $x=-2$. Only the point $x=0$ belongs to the given interval $[-1,2]$.
4. We calculate the value of the function at the point $x=0$ and the values of the function at the ends of the given interval. We have

$$
f(0)=5, \quad f(-1)=0, \quad f(2)=3
$$

By comparing these values, we conclude that the minimum value of the function $m=0$ is achieved at the point $x=-1$, and the maximum value $M=5$ - at the point $x=0$.

Answer. The function takes the minimum value $m=0$ at the point $x=-1$, and the maximum value $M=5$ - at the point $x=0$.

Conditions of the Problems. Find the maximum and minimum values of the function $y=f(x)$ on the interval $[a, b]$.

1. $y=1-\sqrt[3]{x^{2}-2 x}, \quad[0,2]$.
2. $y=2 x-3 \sqrt[3]{x^{2}}, \quad[-1,1]$.
3. $y=x^{2}(x-2)^{2}, \quad[0,2]$.
4. $y=\left(x^{3}-9 x^{2}\right) / 4+6 x-9, \quad[0,4]$.
5. $y=4 x^{2} /\left(3+x^{2}\right), \quad[-1,1]$.
6. $y=1+\sqrt[3]{x^{2}+2 x}, \quad[-2,0]$.
7. $y=2 x^{3}-3 x^{2}-4, \quad[0,2]$.
8. $y=3 \sqrt[3]{(x-3)^{2}}-2 x+6, \quad[2,4]$.
9. $y=\sqrt[3]{x^{2}(x-2)^{2}}, \quad[0,2]$.
10. $y=2-12 x^{2}-8 x^{3}, \quad[-2,0]$.

Answers. 1. $m=y(0)=y(2)=1, M=y(1)=2 . \quad 2 . \quad m=y(-1)=$ $=-5, \quad M=y(0)=0 . \quad 3 . m=y(0)=y(2)=0, \quad M=y(1)=1$. 4. $m=y(0)=-9, \quad M=y(2)=-4 . \quad$ 5. $m=y(0)=0, \quad M=$ $=y(-1)=y(1)=1 . \quad 6 . m=y(-2)=y(1)=1, \quad M=y(-1)=2$. 7. $m=y(1)=-5, \quad M=y(2)=0$. $\quad$ 8. $m=y(3)=0, \quad M=y(2)=5$. 9. $m=y(0)=y(2)=0, \quad M=y(1)=1 . \quad 10 . m=y(-1)=-2$, $M=y(-2)=18$.

## 5.3. Investigation of a Function Using Higher-Order Derivatives

Problem Statement. Investigate the function $y=f(x)$ in the neighborhood of the point $x=a$ using higher-order derivatives.

Solution Plan. Suppose for some $k>1$

$$
f^{\prime}(a)=f^{\prime \prime}(a)=\ldots=f^{(k-1)}(a)=0, \quad f^{(k)}(a) \neq 0
$$

Then if $k$ is an even number, the point $a$ is a point of extremum, specifically a point of maximum if $f^{(k)}(a)0$. If $k$ is an odd number, then the point $a$ is a point of inflection.