Behaviour of a polynomial function

The behaviour of a polynomial function can be studied using a derivative. Since the derivative at a certain point of the function is the slope of the tangent drawn at this point, much can already be deduced from the sign of the derivative.

As the graph of the function increases, all the tangents become ascending lines, the slope is positive. The derivative is positive as the function increases.

When the graph of the function decreases, all the tangents become descending lines, the slope is negative. The derivative is negative when the function decreases.

At points where the graph of the function changes direction, the tangent is parallel to the x-axis and its slope is 0. The function changes direction at the zero points of the derivative.

Below is a graph of the function f and the tangents drawn onto it.

At points where the function changes direction the derivative is zero. Tangents drawn in the image in red. Before the first change of direction, the function is increasing and the tangents drawn there are ascending lines. The figure shows the green tangent at -1.

Between the direction change points -the zero points of the derivative- the values of the function decrease. The tangents drawn to this area are decreasing lines. A blue tangent drawn at x = 0,5.

We can study the behaviour of a function by examining its derivative.

Example 1

When is the function f increasing and when is it decreasing?

We find the derivative and examine its signs

The zero point of the derivative is x = 1. The graph of the derivative is an ascending line

Thus, the derivative has negative values when x < 1 and positive values when x > 1. From this we can deduce the behaviour of the function.

Decreasing, when

Increasing, when

Equality is included during increasing and decreasing. Below is a graph of the function f in blue and a graph of the derivative in green.

Extreme values

The extreme values are the values of the function that are reached at the zero points of the derivative. These points are called extreme points.

Example 2

Find the extreme points and extreme values of the function f.

We find the derivative and then find its zeros

The zeros of the derivative are x = -1 and x = 2. These are extreme points.

The derivative graph is an upward opening parabola. Let's make a table of signs.

Below the signs of the derivative is the behaviour chart of the function. When the derivative is positive, the function increases and the function decreases when the derivative is negative. Point -1 is the zero point of the derivative. The function changes direction and this is the local maximum point. The second zero point of the derivative, point 2, has a local minimum point.

Extreme values of a function

Example 3

Find the extreme values of the function g

We find the derivative and then find its zeros

The derivative has only one zero point x = 2. The graph of the derivative is an upward-opening parabola, so it gets no negative values anywhere. Let's make a behaviour chart of the function.

After the zero point of the derivative, the function continues to increase. The function has no extreme values at all. This is a saddle point.

An absolute extremum

The continuous function gets its maximum and minimum values in a closed interval, either at the endpoints of the interval or at the zero points of the derivative that are in this interval.

Example 4

Find the maximum and minimum value of the function f on an interval [2,4]

Derivative

The zeros of the derivative are x = 1 and x = 3. Of these, only x = 3 belongs to the interval [2,4]. We calculate the values of the function at the endpoints of the interval and at the zero point of the derivative that is in the interval.

With x = 4 the function gets the maximum value of the interval which is 3, and in x = 3 the function gets its minimum value of -1. Below is a graph of the function.

Turn on the subtitles if needed