What is the difference between local and global extrema?

Can you explain the difference between local and global extrema?

Sure! Local extrema are points where a function reaches a maximum or minimum within a small neighborhood, while global extrema are the highest or lowest points over the entire domain of the function.

Can you give me an example of each?

Of course! For the function f(x) = -x^2 + 4, the global maximum is at (0, 4), while local extrema can occur at points like (-2, 0) and (2, 0) if we consider a different function.

How do you find these extrema mathematically?

You can find local extrema by taking the derivative of the function, setting it to zero, and solving for critical points. For global extrema, you also evaluate the function at the endpoints of the interval.

Are there any specific methods to ensure you find global extrema?

Yes! Methods like the Extreme Value Theorem can help ensure you find global extrema by checking critical points and endpoints.