Let's solve these questions step by step.

Question 1:

Find the two closest points among:

• $P_1 = (9, 1, -3)$
• $P_2 = (6, 5, 4)$
• $P_3 = (3, -2, 3)$

The distance between two points $P = (x_1, y_1, z_1)$ and $Q = (x_2, y_2, z_2)$ is given by: $d(P, Q) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$

Compute distances:

1. Distance between $P_1$ and $P_2$: $d(P_1, P_2) = \sqrt{(6-9)^2 + (5-1)^2 + (4-(-3))^2} = \sqrt{(-3)^2 + 4^2 + 7^2} = \sqrt{9 + 16 + 49} = \sqrt{74}$

2. Distance between $P_1$ and $P_3$: $d(P_1, P_3) = \sqrt{(3-9)^2 + ((-2)-1)^2 + (3-(-3))^2} = \sqrt{(-6)^2 + (-3)^2 + 6^2} = \sqrt{36 + 9 + 36} = \sqrt{81} = 9$

3. Distance between $P_2$ and $P_3$: $d(P_2, P_3) = \sqrt{(3-6)^2 + ((-2)-5)^2 + (3-4)^2} = \sqrt{(-3)^2 + (-7)^2 + (-1)^2} = \sqrt{9 + 49 + 1} = \sqrt{59}$

Conclusion:
The closest points are $P_2$ and $P_3$ since $\sqrt{59}$ is smaller than $\sqrt{74}$ and $9$.

Answer: $P_2$ and $P_3$.

Question 2:

Find the $y$-component of the midpoint of:

• Point 1: $(-2, 7, 14)$
• Point 2: $(8, -2, 5)$

The midpoint formula is: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right)$

For the $y$-component: $y = \frac{y_1 + y_2}{2} = \frac{7 + (-2)}{2} = \frac{5}{2} = 2.5$

Answer: $y = 2.5$.

Question 3:

Find the vertex of a function.
To provide a specific answer, the function itself (e.g., a quadratic or cubic equation) is required. Without the function details, the correct answer cannot be determined. Could you provide the function?

Would you like me to go over these solutions in more detail or clarify anything?

Here are 5 related questions to extend your understanding:

1. How do we generalize the midpoint formula to $n$-dimensional space?
2. What is the geometric meaning of the distance formula in 3D?
3. Can the vertex formula be used for functions other than quadratics?
4. How do transformations (like translations) affect the location of a vertex?
5. What happens to the midpoint formula if one of the components is undefined or infinite?

Tip: Always double-check the function or equation provided to ensure you're applying the right formulas.