Based on the uploaded images, I will interpret the mathematical statements and verify their correctness regarding the function $y = x^3$.

Key properties of $y = x^3$:

1. First derivative: $y'(x) = 3x^2$

   • $y'(x) = 0$ at $x = 0$, which is a critical point.

2. Second derivative: $y''(x) = 6x$

   • $y''(x) > 0$ for $x > 0$ (concave up).
   • $y''(x) < 0$ for $x < 0$ (concave down).
   • $y''(0) = 0$, so $x = 0$ is a point of inflection.

3. Behavior at $x = 0$:

   • $y = x^3$ is continuous and differentiable everywhere, including at $x = 0$.

Analysis of the statements:

1. First statement: Since $y''(x) > 0$ on $(0, +\infty)$, the function is strictly concave up on this interval. The statement says "strictly concave down," which is incorrect.

2. Second statement: The function $y = x^3$ is continuous at $x = 0$. This statement is correct.

3. Third statement: On $(-\infty, 0)$, $y''(x) < 0$, so the function is concave down on this interval. This statement matches, so it is correct.

4. Fourth statement: At $x = 0$, the second derivative $y''(0) = 0$, and $x = 0$ is not a critical point of the second kind. The statement is incorrect.

5. Fifth statement: The second derivative $y''(0) = 0$, which is correct.

6. Sixth statement: On $(-\infty, 0)$, $y''(x) < 0$, so it is concave down, not concave up. This statement is incorrect.

7. Seventh statement: The first derivative $y'(0) = 0$, which exists. The statement is correct.

8. Eighth statement: The second derivative $y''(0) = 0$, and it exists. The statement claiming non-existence is incorrect.

Correct Statements:

1. The function is continuous at $x = 0$.
2. The function is concave down on $(-\infty, 0)$.
3. $y'(0)$ exists and equals $0$.
4. $y''(0)$ exists and equals $0$.

Would you like detailed explanations for each statement or assistance with additional concepts? Here are five questions you might explore further:

1. What defines a point of inflection for a function like $y = x^3$?
2. How does the second derivative test classify critical points?
3. Can a function be both concave up and concave down at different intervals?
4. Why is continuity important for analyzing derivatives at a point?
5. How does $y = x^3$ compare to higher-degree odd functions?

Tip: Always examine the behavior of derivatives to confirm concavity and points of inflection.