The First Derivative Test is a helpful method for understanding how functions behave. It helps us find important points and whether these points are high or low on a graph. This test is super useful in calculus because it shows when a function goes up or down, and this information is key for drawing graphs and finding local highest and lowest points.

What Are Critical Points?

Critical points are special places where the first derivative of a function, written as ( f'(x) ), is either zero or doesn’t exist. These points are where the behavior of the function can change from going up to going down or the other way around.

To find these critical points, we usually set the first derivative equal to zero:

[ f'(x) = 0 ]

Let’s look at an example with the function ( f(x) = x^3 - 3x^2 + 4 ). First, we find the derivative:

[ f'(x) = 3x^2 - 6x ]

Setting this equal to zero gives us:

[ 3x^2 - 6x = 0 ] [ 3x(x - 2) = 0 ]

So, the critical points are ( x = 0 ) and ( x = 2 ).

Using the First Derivative Test

Now, let's talk about what we do with these critical points! The First Derivative Test helps us figure out what kind of points these are. Here’s the simple process:

1. Pick Test Points: Choose points in the sections created by the critical points. For our function, we have the sections:

   • ( (-\infty, 0) )
   • ( (0, 2) )
   • ( (2, \infty) )

2. Evaluate the Sign of ( f'(x) ): Choose test points from each section, like ( -1 ), ( 1 ), and ( 3 ):

   • For ( x = -1 ): [ f'(-1) = 3(-1)^2 - 6(-1) = 9 \quad \text{(positive, function is increasing)} ]
   • For ( x = 1 ): [ f'(1) = 3(1)^2 - 6(1) = -3 \quad \text{(negative, function is decreasing)} ]
   • For ( x = 3 ): [ f'(3) = 3(3)^2 - 6(3) = 9 \quad \text{(positive, function is increasing)} ]

3. Draw Conclusions: What can we learn from this?

   • At ( x = 0 ), the function goes from increasing to decreasing. This means it’s a local maximum.
   • At ( x = 2 ), the function goes from decreasing to increasing. This means it’s a local minimum.

Conclusion

The First Derivative Test gives us important information about how a function behaves near its critical points. By knowing where a function goes up or down, we can make accurate graphs and solve problems more easily. This test helps us find local maximum and minimum points, which is really useful in calculus and in daily life!