Local extrema refer to the points in a function where the function reaches either a local maximum or a local minimum. A local maximum is a point where the function's value is higher than all nearby points, and a local minimum is where the function's value is lower than all nearby points.

Definitions:

1. Local Maximum: A function $f(x)$ has a local maximum at $x = c$ if $f(c) \geq f(x)$ for all $x$ in a neighborhood around $c$.
2. Local Minimum: A function $f(x)$ has a local minimum at $x = c$ if $f(c) \leq f(x)$ for all $x$ in a neighborhood around $c$.

How to Find Local Extrema:

1. Take the Derivative: Compute the first derivative $f'(x)$ of the function.
2. Find Critical Points: Set $f'(x) = 0$ or find where $f'(x)$ is undefined. These are the critical points.
3. Second Derivative Test (Optional for confirmation):
   • If $f''(x) > 0$, there is a local minimum at that point.
   • If $f''(x) < 0$, there is a local maximum at that point.
4. First Derivative Test: Alternatively, check the sign of the first derivative around the critical points to confirm whether they are maxima or minima.

Example:

Let’s take $f(x) = x^3 - 3x^2 + 4$.

1. $f'(x) = 3x^2 - 6x$.
2. Set $f'(x) = 0$ to find critical points: $3x(x - 2) = 0$, so $x = 0$ and $x = 2$.
3. Compute the second derivative: $f''(x) = 6x - 6$.
   • At $x = 0$, $f''(0) = -6$, indicating a local maximum.
   • At $x = 2$, $f''(2) = 6$, indicating a local minimum.

Would you like to see a more detailed explanation or explore specific types of problems related to local extrema?

Here are 5 questions you could explore further:

1. How do we handle functions with multiple local extrema?
2. What is the role of the second derivative in classifying critical points?
3. Can a critical point occur where the derivative is undefined?
4. How do local extrema differ in polynomial vs trigonometric functions?
5. What happens when both the first and second derivatives are zero at a critical point?

Tip: Always check the behavior of the function at endpoints if you're working within a closed interval, as extrema can occur there too!