Let's analyze the statements one by one to determine which are correct.

1. $y = \ln x$ is bounded on $(0; 10)$.

• The function $\ln x$ is unbounded as $x \to 0^+$, since $\ln x \to -\infty$. However, as $x \to 10$, $\ln x$ remains finite. Therefore, $\ln x$ is unbounded from below but bounded from above on $(0, 10)$.
• This statement is incorrect.

2. $y = \frac{1}{\sqrt{2\pi}} e^{-x^2 / 2}$ is bounded on $(-\infty; +\infty)$.

• This function is a Gaussian (normal distribution) function. It is always non-negative and tends to 0 as $x \to \infty$ or $x \to -\infty$. The maximum value is $\frac{1}{\sqrt{2\pi}}$, occurring at $x = 0$.
• This function is indeed bounded for all real numbers.
• This statement is correct.

3. $y = \sin x$ is unbounded on $(0; \pi)$.

• The sine function oscillates between $-1$ and $1$. On the interval $(0; \pi)$, $\sin x$ only takes values between $0$ and $1$, so it is bounded.
• This statement is incorrect.

4. $y = \frac{1}{x}$ is bounded from below on $(0; 10)$.

• The function $\frac{1}{x}$ tends to $+\infty$ as $x \to 0^+$. However, it decreases as $x$ increases and is bounded from below by $\frac{1}{10}$.
• This statement is incorrect because it is not bounded from below (it becomes arbitrarily large as $x$ approaches 0).

5. $y = \ln x$ is bounded from above on $(0; 10)$.

• We have already established that $\ln x$ tends to $-\infty$ as $x \to 0^+$, but it is bounded from above since $\ln 10$ is finite.
• This statement is correct.

Correct answers:

• Statement 2: $y = \frac{1}{\sqrt{2\pi}} e^{-x^2 / 2}$ is bounded on $(-\infty; +\infty)$.
• Statement 5: $y = \ln x$ is bounded from above on $(0; 10)$.

Do you want more details on any of the explanations?

Here are 5 related questions:

1. What is the behavior of the logarithmic function near zero?
2. How do we determine if a function is bounded?
3. What are the properties of Gaussian functions like $\frac{1}{\sqrt{2\pi}} e^{-x^2 / 2}$?
4. Can a function be unbounded in one direction but bounded in another?
5. How does the sine function behave on different intervals?

Tip: When checking if a function is bounded, always analyze its limits at the boundaries of the given domain.