To find the local maximum and local minimum values of a function, you generally follow these steps:

1. Find the first derivative of the function $f'(x)$.
2. Set the first derivative equal to zero to find the critical points, i.e., solve $f'(x) = 0$.
3. Determine the nature of each critical point:
   • Use the second derivative test by finding $f''(x)$:
     • If $f''(x) > 0$, the function has a local minimum at that point.
     • If $f''(x) < 0$, the function has a local maximum at that point.
   • Alternatively, you can use the first derivative test to check the sign changes of $f'(x)$ around the critical points.
4. Plug critical points into the original function to find the corresponding function values, which are the local maximum and local minimum values.

If you provide the specific function, I can perform these steps and list the local maximum and minimum values for you.

Would you like to proceed with a specific function?

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Here are five related questions you may find useful:

1. What are the global maximum and minimum values of a function?
2. How does the first derivative test work in determining local extrema?
3. What role does concavity play in identifying local maxima and minima?
4. Can a function have more than one local maximum or minimum?
5. How can graphing help in visualizing local extrema?

Tip: Always check the endpoints of a domain when looking for extrema on a closed interval.