Stationary Points, Increasing/Decreasing Functions

INTRODUCTION TO THE BEHAVIOR OF FUNCTIONS

To understand the behavior of a function and determine its key properties, we use the analysis of its derivative. One of the most important properties we discover this way is where the function is increasing, where it is decreasing, and where it has its extreme values. So-called stationary points, which are determined using the first derivative, serve this purpose.

DEFINITION OF STATIONARY POINTS

Let 'f' be a differentiable function. A point x₀ is stationary if f′(x₀) = 0. At such a point, the tangent to the graph of the function changes from increasing to decreasing, or vice versa – or it remains horizontal.

Stationary points can be:

• A local minimum, if the function at this point reaches the smallest value in some neighborhood.
• A local maximum, if it reaches the largest value in a neighborhood.
• A point of inflection (or saddle point), if the direction of the function does not change (e.g., it remains increasing), but the shape of the graph (concavity) changes around this point. (Note: Inflection points are more formally identified using the second derivative or changes in concavity, but a horizontal tangent at an inflection point means f'(x)=0).

INCREASING AND DECREASING NATURE OF A FUNCTION

Using the derivative, we can also determine the intervals where a function is increasing or decreasing:

• If f′(x) > 0 for all x in some interval, then the function is increasing in that interval.
• If f′(x) < 0 for all x in some interval, the function is decreasing in that interval.
• If f′(x) = 0 at individual points, these are potential stationary points.

This information is often collected in a sign table for the derivative, which helps in sketching the graph of the function.

EXAMPLE: FUNCTION F(X) = X³ – 3X² + 2

First, let's find the derivative:
f′(x) = 3x² – 6x.

Next, we solve the equation f′(x) = 0:
3x² – 6x = 0 → x(3x – 6) = 0 → x = 0 or x = 2.
These are the stationary points.

Then, we check the sign of the derivative in the intervals defined by these points:

• For x < 0: Choose a test value, e.g., x = -1. f′(-1) = 3(-1)² – 6(-1) = 3 + 6 = 9 > 0 → the function is increasing.
• For 0 < x < 2: Choose a test value, e.g., x = 1. f′(1) = 3(1)² – 6(1) = 3 – 6 = -3 < 0 → the function is decreasing.
• For x > 2: Choose a test value, e.g., x = 3. f′(3) = 3(3)² – 6(3) = 27 – 18 = 9 > 0 → the function is increasing again.

From this, we conclude that at x = 0 there is a local maximum, and at x = 2 there is a local minimum.

CONCLUSION

Stationary points are key in the analysis of functions, as they mark places where the direction of the graph changes. Using the derivative, we determine where a function is increasing or decreasing, which allows for a precise understanding of its behavior and course.