Aus Online Mathematik Brückenkurs 2

Theorie

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Increasing and decreasing

Informally, a function is increasing if its graph slopes upwards and decreasing if its graph slopes downwards.

The formal 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 of a function being constant on an interval, one 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 or 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, however, that this works only one way round. It is entirely possible for a function to be strictly increasing or decreasing on some interval, and for there to be a point, or perhaps more than one, within that interval at which the derivative is zero. As long as any points where the derivative is zero are isolated (that is, provided the derivative isn't zero anywhere close to such points, but only at them), then the function can be strictly increasing or decreasing; problems only arise if the gradient is zero, and the function therefore constant, over some interval.

Stationary points

Points where \displaystyle f^{\,\prime}(x) = 0 are known as stationary points (the term "critical points" is also sometimes used). They 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.
• Stationary point of inflexion 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 maxima and minima.

Points of inflexion

A point of inflexion is a point where the direction of curvature of a graph changes: where, having curved upwards, it begins curving downwards, or vice versa. In other words, it is a local maximum or minimum for the gradient.

One way to think about points of inflexion is to imagine driving a car along a road shaped like the curve; a point of inflexion is any point at which your steering wheel is exactly centred and you are, for that instant, steering neither right nor left.

At a point of inflexion, the curve's direction of curvature changes; the curve on the left has a stationary point of inflexion in x = 0, where the gradient is zero, but as the other two curves show, not all points of inflexion are stationary points.

We will not study points of inflexion in depth in this section, except to note that a point of inflexion need not necessarily also be a stationary point (though the two can coincide, in which case we would often call the point concerned a "stationary point of inflexion").

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

Table of signs

By studying the derivative sign (+, - or 0), we can therefore 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 (together with any where the gradient is undefined) and then calculates the sign of the derivative on either side 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.

Maxima and minima (extrema)

A point at which a function takes on its largest or smallest value in comparison with its immediate surroundings is called a local maximum or local minimum (often abbreviated to max and min). Local maxima and minima are together known as extrema.

An extremum may occur in one of three ways:

• At a stationary 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 the extrema 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 locate and classify all of the extrema.
5. Calculate the value of the function for all the extrema and at any endpoints.

Global min / max

A function has aglobal maximum at a point if its value there is greater than, or at least equal to, its value at any other point where it is defined; similarly, a global minimum is a point where the function's value is less than, or at most equal to, its value anywhere else.

To determine a function's global max or min one must therefore find all the extrema and calculate the values of the function at them. If the function is defined on an interval with endpoints, one must of course also examine its value at these points.

Note that a function need not have a global max or a global min, even if it has several local extrema.

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.

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 stationary 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 sign of the second derivative can show if the first order derivative is increasing or decreasing. This can , among other things, be used to find out whether a given extremum is a maximum or minimum.

If the function \displaystyle f(x) has a stationary 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.

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 stationary 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, and further investigation is required, for example by means of a table of signs. Note in particular that \displaystyle f^{\,\prime\prime}(a)=0 does not imply that the point is a stationary point of inflexion (although \displaystyle f^{\,\prime\prime}(a)=0 at all points of inflexion, it can also be zero elsewhere, including at maxima and minima).