Understanding critical points is really important when looking at how a graph of a function behaves.

A critical point happens at a value of $x$ where the derivative $f'(x)$ is either zero or undefined. These points can show us where the graph might have a peak, a valley, or change its bending shape.

Types of Critical Points:

1. Local Maxima: This is where the function goes from increasing (getting higher) to decreasing (getting lower).

2. Local Minima: This is where the function goes from decreasing to increasing.

3. Points of Inflection: This is where the curve changes its bending but isn't a peak or a valley.

First Derivative Test:

The First Derivative Test helps us figure out what type of critical point we have:

• If $f'(x)$ changes from positive (going up) to negative (going down) at a critical point $c$, then $f(c)$ is a local maximum.

• If $f'(x)$ changes from negative to positive at a critical point $c$, then $f(c)$ is a local minimum.

• If $f'(x)$ does not change at $c$, then it is neither a maximum nor a minimum.

Example:

Let's look at the function $f(x) = x^3 - 3x^2 + 4$.

1. First, we find the derivative: $f'(x) = 3x^2 - 6$.

2. Next, we set $f'(x) = 0$ to find critical points: $3x^2 - 6 = 0 \implies x^2 = 2 \implies x = \sqrt{2}, -\sqrt{2}$.

By finding these critical points and using the First Derivative Test, we can understand how the graph behaves. This makes it easier to see how the function changes and what it does overall.