The problem is asking to determine the intervals where the given functions are increasing or decreasing. To solve these, we need to compute the derivative of each function, set it equal to zero to find critical points, and test intervals around these points.

I’ll work through one example in detail. Let’s begin with 1(a):

Function: $f(x) = x^2 - 7x + 10$

1. Find the derivative: $f'(x) = 2x - 7$

2. Set the derivative to zero to find critical points: $2x - 7 = 0 \quad \Rightarrow \quad x = \frac{7}{2}$

3. Determine intervals:

   • Test values on either side of $x = \frac{7}{2}$:
     • For $x < \frac{7}{2}$ (e.g., $x = 0$): $f'(0) = 2(0) - 7 = -7$ (negative, decreasing).
     • For $x > \frac{7}{2}$ (e.g., $x = 4$): $f'(4) = 2(4) - 7 = 1$ (positive, increasing).

4. Conclusion:

   • $f(x)$ is decreasing on $(-\infty, \frac{7}{2})$
   • $f(x)$ is increasing on $(\frac{7}{2}, \infty)$

Would you like me to proceed with other parts of this question? If yes, let me know if you'd like a detailed explanation for all parts or only specific ones.

Here are five related questions:

1. What is the relationship between the derivative and the monotonicity of a function?
2. How do critical points help identify increasing and decreasing intervals?
3. What happens when the derivative of a function does not change sign at a critical point?
4. How do you handle rational functions (e.g., 4(a)) when finding intervals?
5. What does the second derivative reveal about the nature of critical points?

Tip: Always sketch the graph of the derivative to better understand the function’s behavior!