The uploaded image contains problems related to derivatives and the intervals of increase or decrease for quadratic functions. Let me break down the structure and solution provided in the example problem and work through it.

Given Problem:

The general quadratic function $y = ax^2 + bx + c$ (where $a \neq 0$) is analyzed for intervals of increase and decrease based on the derivative.

Example (in blue box):

1. The function $y = x^2 - 6x$ is given.
2. The derivative is calculated as: $f'(x) = 2x - 6$
3. Setting $f'(x) = 0$: $2x - 6 = 0 \quad \Rightarrow \quad x = 3$ This is the critical point.
4. The derivative $f'(x)$ is tested on intervals $(-\infty, 3)$ and $(3, \infty)$:
   • For $x < 3$: $f'(x) < 0$, so the function is decreasing.
   • For $x > 3$: $f'(x) > 0$, so the function is increasing.

Thus, the function is increasing on $(3, +\infty)$ and decreasing on $(-\infty, 3)$.

For the Other Problems:

We follow the same steps for the functions:

1. Find the derivative of each function $f(x)$.
2. Solve $f'(x) = 0$ to find critical points.
3. Analyze the sign of $f'(x)$ on intervals determined by the critical points.
4. Determine where the function increases ($f'(x) > 0$) or decreases ($f'(x) < 0$).

Would you like me to solve one of the provided sub-problems (a, b, c, or d) step by step? Let me know! 😊

5 Follow-Up Questions:

1. How do you determine the critical points of a function using its derivative?
2. Why does the sign of $f'(x)$ determine whether a function increases or decreases?
3. What happens if a critical point occurs at $x = 0$? How would it affect intervals?
4. How would you apply these concepts to non-quadratic functions?
5. How do you check for concavity (increasing vs. decreasing second derivative)?

Tip: Always plot the function or derivative graph if you’re unsure about interval behavior—it provides a clear visual confirmation.