Let's solve these questions step by step in brief.

Question 12:

Given: $f(x) = kx^3 + 9x^2$ and there's an inflection point at $x = -1$.

• The inflection point occurs where the second derivative equals zero and changes sign.
• First derivative: $f'(x) = 3kx^2 + 18x$
• Second derivative: $f''(x) = 6kx + 18$
• Setting $f''(-1) = 0$: $6k(-1) + 18 = 0 \quad \Rightarrow \quad -6k + 18 = 0 \quad \Rightarrow \quad k = 3$

Answer: $\boxed{3}$

Question 13:

Given: $y = x^3 + ax^2 + bx$ has an inflection point at $(3, -9)$.

• For inflection points, the second derivative must be zero.
• First derivative: $y' = 3x^2 + 2ax + b$
• Second derivative: $y'' = 6x + 2a$
• At $x = 3$, $y''(3) = 0$: $6(3) + 2a = 0 \quad \Rightarrow \quad 18 + 2a = 0 \quad \Rightarrow \quad a = -9$
• Substituting into $y(3) = -9$: $27 + 9(-9) + 3b = -9 \quad \Rightarrow \quad 27 - 81 + 3b = -9 \quad \Rightarrow \quad 3b = 45 \quad \Rightarrow \quad b = 15$
• Thus, $a + b = -9 + 15 = 6$.

Answer: $\boxed{6}$

Question 14:

Points separating concavity up and down are called inflection points (where the second derivative changes sign).

Answer: $\boxed{\text{inflection}}$

Question 15:

Given: $f(x) = x^4$.

• $f'(x) = 4x^3$, and $f''(x) = 12x^2$.
• At $x = 0$, $f'(0) = 0$ and $f''(0) = 0$, which is an inflection point since the concavity changes from up to down at $(0, 0)$.

Answer: $\boxed{\text{inflection point}}$

Question 16:

If the curve of a function has an inflection point, the maximum number of points where it intersects a straight line equals 3, because an inflection point indicates a cubic or higher degree polynomial which can intersect a straight line at most three times.

Answer: $\boxed{3}$

Would you like more details on any of the steps? Here are five related questions you might find useful:

1. What is the significance of the second derivative in determining concavity?
2. How do we confirm the nature of an inflection point?
3. Can a cubic function have more than one inflection point?
4. How is the number of intersections between a polynomial and a line determined?
5. What are the key features of a function’s critical points?

Tip: Always verify the second derivative changes sign to confirm an inflection point, not just that it equals zero!