Got it! The image contains formulas and rules about analyzing functions using the first and second derivative tests, as well as determining critical points, local extrema, and inflection points. Here's a summary of the rules:

Key Concepts:

1. First Derivative Test:

   • Critical points occur where $f'(x) = 0$.
   • These points help identify potential local maxima, minima, or saddle points.
   • Critical points are expressed as $(x, f(x))$.

2. Second Derivative Test:

   • Helps classify the nature of critical points.
   • If $f''(x) < 0$: Local maximum.
   • If $f''(x) > 0$: Local minimum.
   • If $f''(x) = 0$ or $f''(x)$ is undefined: Cannot determine the nature; further analysis needed.

3. Inflection Points:

   • These occur where $f''(x) = 0$ and the sign of $f''(x)$ changes across the point.
   • To confirm an inflection point: $\text{Check if } \text{sign}(f''(x - \epsilon)) \neq \text{sign}(f''(x + \epsilon)).$

Would you like me to work on specific problems, provide detailed examples, or clarify any of the points?

Here are 5 relative questions to consider:

1. How do you compute critical points for a given polynomial?
2. What does it mean if $f'(x) = 0$ but $f''(x) = 0$ at a critical point?
3. Can a point where $f'(x) \neq 0$ still be an inflection point?
4. How does the concavity of a function relate to its second derivative?
5. What’s the procedure for solving a higher-order polynomial to find local extrema?

Tip: Always sketch the derivative curve when unsure—it visually clarifies increasing, decreasing, and concavity trends!