Aus Online Mathematik Brückenkurs 2

Theory

Exercises

Increasing and decreasing

The concepts of increasing and decreasing may seem obvious when talking about mathematical functions, if the function is increasing then the graph slopes upwards and if it is decreasing it slopes downwards.

The mathematical definitions are as follows:

A function is increasing in an interval if for all \displaystyle x_1 and \displaystyle x_2 within the interval

\displaystyle x_1 < x_2\quad\Rightarrow\quad f(x_1) \le f(x_2)\,\mbox{.}

A function is decreasing in an interval if for all \displaystyle x_1 and \displaystyle x_2 within the interval

\displaystyle x_1 < x_2\quad\Rightarrow\quad f(x_1) \ge f(x_2)\,\mbox{.}

In everyday language the definition says, for example, that for an increasing function for any x-value to the right on the x-axis, the value of the function is at least as large as it is for any x-value to the left. Please note that this definition means that a function can be constant in a interval, and still be considered to be increasing or decreasing. A function that is constant throughout an interval, according to the definition, is both increasing and decreasing.

If one wants to exclude the possibility that an increasing / decreasing function is constant in an interval one rather talks of strictly increasing and strictly decreasing functions:

A function is strictly increasing in an interval if for all \displaystyle x_1 and \displaystyle x_2 within the interval

\displaystyle x_1 < x_2\quad\Rightarrow\quad f(x_1) < f(x_2)\,\mbox{.}

A function is strictly decreasing in an interval if for \displaystyle x_1 och \displaystyle x_2 within the interval

\displaystyle x_1 < x_2\quad\Rightarrow\quad f(x_1) > f(x_2)\,\mbox{.}

(A strictly increasing / decreasing function cannot be constant in any part of the interval.)

The derivative may of course be used to examine whether a function is increasing or decreasing. We have that

\displaystyle \begin{align*} f^{\,\prime}(x) > 0 \quad&\Rightarrow \quad f(x) \text{ is (strictly) increasing,}\\ f^{\,\prime}(x) < 0 \quad&\Rightarrow \quad f(x) \text{ is (strictly) decreasing.} \end{align*}

Note that even isolated points where \displaystyle f^{\,\prime}(x) = 0 can be part of a strictly increasing or decreasing interval.

Critical points

Points where \displaystyle f^{\,\prime}(x) = 0 are known as critical (or stationary) points and are usually one of three kinds:

• Local maximum with \displaystyle f^{\,\prime}(x) > 0 to the left, and \displaystyle f^{\,\prime}(x) < 0 to the right of the point.
• Local minimum with \displaystyle f^{\,\prime}(x) < 0 to the left, and \displaystyle f^{\,\prime}(x) > 0 to the right of the point.
• Point of inflection with \displaystyle f^{\,\prime}(x) < 0 or \displaystyle f^{\,\prime}(x) > 0 on both sides of the point.

Note that a point may be a local maximum or minimum without \displaystyle f^{\,\prime}(x) = 0; learn more about this in the section on maximums and minimums.

1.3 - Figure - The graph of f(x) = x³ - x⁵

The function in the above figure has a local minimum for \displaystyle x = -2, a point of inflection for \displaystyle x = 0 and a local maximum for \displaystyle x = 2.

Table of signs

By studying the derivative sign (+, - or 0), we therefore can obtain a good idea of the curve's appearance.

One creates a so called table of signs. One first determines the x-values where \displaystyle f^{\,\prime}(x) =0 and then calculates the sign of the derivative on both sides of these points. With the help of some other "backup" points on the curve and using the table of signs one usually can obtain a satisfactory sketch of the curve.

Maximums and minimums (extreme points)

A point in which a function takes on its largest or smallest value in comparison with its surroundings is called a local maximum or local minimum (often abbreviated to max and min). A joint name is extreme point or just extreme.

A extreme may occur in one of three ways:

• At a critical point (where \displaystyle f^{\,\prime}(x)=0\,).
• At a point where the derivative does not exist (known as a singular point).
• At an endpoint to the interval where the function is defined.

When one is looking for extreme points of a function one must discover and examine all possible candidates for these points. An appropriate working procedures is:

1. Differentiate the function.
2. Check to see if there are any points where \displaystyle f^{\,\prime}(x) is not defined.
3. Determine all points where \displaystyle f^{\,\prime}(x) = 0.
4. Make a table of signs to get all of the extreme points.
5. Calculate the value of the function for all the extreme points and at any endpoints.

Absolute min / max

A function has an absolute (or global) maximum (minimum) at a point if the value of the function is not greater (less) in any other point in the interval where the function is defined. This often is called the functions largest (least) value.

To determine a functions absolute max or min one must therefore find all the extreme points and calculate the values of the function at them. If the interval where the function is defined has endpoints, one must of course examine the function value at these points.

Note that an function may lack both an absolute max and an absolute min. Note also that a function can have several local extreme points without having a global max or min.

In applications, circumstances often dictate that a function has a limited interval where it is defined, i.e. one only studies part of the graph of the function. One must therefore be careful in case the global max or min is at an endpoint of the interval.

1.3 - Figur - Funktion med lokala och globala extrempunkter

The above function is only of interest in the interval \displaystyle a\le x \le e. We see that the minimum value of the function in this interval occurs at the critical point, \displaystyle x=b, while the maximim value is found at the endpoint \displaystyle x=e.

The second derivative

The sign of the derivative of a function gives us information about whether the function is increasing or decreasing. Similarly, the second order derivatives sign can show if the first order derivative is increasing or decreasing. This can , among other things, be used to find out whether a given extreme point is a maximum or minimum.

If the function \displaystyle f(x) has a critical point at \displaystyle x=a where \displaystyle f^{\,\prime\prime}(a)<0, then

1. The derivative \displaystyle f^{\,\prime}(x) is strictly decreasing in some interval surrounding \displaystyle x=a.
2. Since \displaystyle f^{\,\prime}(a)=0 then \displaystyle f^{\,\prime}(x)>0 to the left of \displaystyle x=a and \displaystyle f^{\,\prime}(x)<0 to the right of \displaystyle x=a.
3. This means that the function \displaystyle f(x) has a local maximum at \displaystyle x=a.

1.3 - Figure - The tangent of a function with negative second order derivative

If the derivative is positive to the left of x = a and negative to the right of x = a the function has a local maximum at x = a.

If the function \displaystyle f(x) has a critical point at \displaystyle x=a where \displaystyle f^{\,\prime\prime}(a)>0, then

1. The derivative \displaystyle f^{\,\prime}(x) is strictly increasing in some interval around \displaystyle x=a.
2. Since \displaystyle f^{\,\prime}(a)=0 then \displaystyle f^{\,\prime}(x)<0 to the left of \displaystyle x=a and \displaystyle f^{\,\prime}(x)>0 to the right of \displaystyle x=a.
3. This means that the function \displaystyle f(x) has a local minimum at \displaystyle x=a.

1.3 - Figure - The tangent of a function with positive second order derivative

If the derivative is negative to the left of x = a and positive to the right of x = a the function has a local minimum at x = a.

If \displaystyle f^{\,\prime\prime}(a)=0, no information can be deduced, but further investigation is required, for example by means of a table of signs.