To understand how a function behaves near important points, we can use a method called the First Derivative Test. Let's break it down step by step.

First, we need to find critical points.

What are critical points?

They happen where the first derivative, which we’ll call $f'(x)$, is either zero or doesn’t exist.

These points help us find possible highs (local maxima) and lows (local minima) of the function.

After finding these critical points, we look at the areas on both sides of each point.

To do this, we pick test points in those areas and check the sign (positive or negative) of $f'(x)$ at these points:

1. Positive Derivative:

   • If $f'(x) > 0$ on the left side of a critical point and $f'(x) < 0$ on the right side, it means the function is going up before the point and going down after it.
   • This tells us the critical point is a local maximum.

2. Negative Derivative:

   • On the other hand, if $f'(x) < 0$ before the critical point and $f'(x) > 0$ after, the function is going down first and then going up.
   • This means the critical point is a local minimum.

3. No Change in Sign:

   • If the sign of $f'(x)$ doesn’t change at all (it stays positive or negative on both sides), then the critical point is neither a maximum nor a minimum.
   • We call this a horizontal inflection point.

By using the First Derivative Test, we can find out whether a critical point is a high point, a low point, or something else.

This information helps us draw the graph and understand how the function behaves overall.