Critical points are important when we analyze graphs of functions. They help us find places where the function might have its highest points, lowest points, or changes direction.

Why They Matter:

1. Local Extrema: Critical points show us where the function could reach its highest or lowest values in a small area. This is very helpful when we want to find the best possible solution to a problem.

2. Graph Shape: These points tell us how the graph is acting around them. They can show if the graph is going up or down.

Example:

Take the function $f(x) = x^3 - 3x^2 + 2$.

To find its critical points, we look for when its derivative, $f'(x) = 3x^2 - 6$, is equal to zero.

When we solve this, we get $x = 0$ and $x = 2$.

These points help us see where the graph has its peaks and valleys.