Finding the highest or lowest points of a graph (known as extrema) can feel tricky at first. But don't worry! Once you learn the key steps, it becomes much easier. Here’s how I learned to do it in my AP Calculus class:

Step 1: Understand the Problem

First, make sure you really understand the function you’re looking at.

• Is it continuous? This means there are no breaks or jumps in the graph.
• What is the domain? This is the set of numbers the function can work with.

These details matter because they help you figure out where you might find the highest or lowest points.

Step 2: Find Critical Points

To find local extrema, you need to calculate the derivative of your function. We call this $f'(x)$.

Critical points are where:

• $f'(x) = 0$ (this means the slope is flat).
• $f'(x)$ does not exist.

Solve the equation $f'(x) = 0$ to find these points. Also, check the places where $f'(x)$ doesn't exist because they could also be important points.

Step 3: Use the Second Derivative Test

After finding your critical points, look at the second derivative, $f''(x)$. This will help you figure out the type of each critical point:

• If $f''(x) > 0$, you have a local minimum (the lowest point nearby).
• If $f''(x) < 0$, you have a local maximum (the highest point nearby).
• If $f''(x) = 0$, you can't tell just yet, and you might need more analysis.

Step 4: Evaluate Endpoints

To find global extrema, look at the endpoints of your function’s domain.

Calculate the function’s value at these endpoints and also at each critical point you found before.

Step 5: Compare Values

Now you have a list of values from both the critical points and the endpoints.

• The global maximum is the biggest value.
• The global minimum is the smallest value.

Summary

In short, here are the steps to follow:

1. Understand the function's domain.
2. Find critical points using the first derivative.
3. Use the second derivative test to check local extrema.
4. Evaluate the function at endpoints.
5. Compare all values to find the global extrema.

Following these steps will help you tackle optimization problems and find extrema with confidence. This makes it a handy skill to have in your math toolbox!