To find out where a function is going up or down, we can use something called the First Derivative Test. This method is really useful, especially when we look at critical points.

Critical points are where the first derivative is either zero or doesn't exist.

Step 1: Find the Derivative

First, let's pick a function, which we’ll call $f(x)$. The first thing we need to do is find its derivative, noted as $f'(x)$. The derivative tells us how the function is changing.

Example:

Take the function $f(x) = x^3 - 3x^2 + 4$.

The first derivative for this function is:

$f'(x) = 3x^2 - 6x.$

Step 2: Identify Critical Points

Next, we set the derivative equal to zero to find our critical points:

$3x^2 - 6x = 0.$

Factoring out what’s common gives us:

$3x(x - 2) = 0.$

So, the critical points we find are $x = 0$ and $x = 2$.

Step 3: Test Intervals Around the Critical Points

Now we will check the intervals made by these critical points. The intervals we have are $(-\infty, 0)$, $(0, 2)$, and $(2, \infty)$.

We will pick test numbers from each interval to see if $f'(x)$ is positive (the function is increasing) or negative (the function is decreasing):

• Interval $(-\infty, 0)$: Let's pick $x = -1$.

  $f'(-1) = 3(-1)^2 - 6(-1) = 3 + 6 = 9 > 0$

  (This means the function is increasing here)

• Interval $(0, 2)$: Now we pick $x = 1$.

  $f'(1) = 3(1)^2 - 6(1) = 3 - 6 = -3 < 0$

  (This means the function is decreasing here)

• Interval $(2, \infty)$: Lastly, we pick $x = 3$.

  $f'(3) = 3(3)^2 - 6(3) = 27 - 18 = 9 > 0$

  (This means the function is increasing here)

Step 4: State the Increasing and Decreasing Intervals

Based on our tests, we find:

• The function is increasing in the intervals $(-\infty, 0)$ and $(2, \infty)$.
• The function is decreasing in the interval $(0, 2)$.

In summary, using the First Derivative Test helps us understand how a function behaves around its critical points. This gives us important information about where the function is going up or down. It's a basic but essential idea for grasping how a function's graph looks!