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Find the average rate of change of f ( x ) = x 2 + 2 x 8 on the interval [ 5 , a ] .

a + 7

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Using a graph to determine where a function is increasing, decreasing, or constant

As part of exploring how functions change, we can identify intervals over which the function is changing in specific ways. We say that a function is increasing on an interval if the function values increase as the input values increase within that interval. Similarly, a function is decreasing on an interval if the function values decrease as the input values increase over that interval. The average rate of change of an increasing function is positive, and the average rate of change of a decreasing function is negative. [link] shows examples of increasing and decreasing intervals on a function.

Graph of a polynomial that shows the increasing and decreasing intervals and local maximum and minimum.
The function f ( x ) = x 3 12 x is increasing on ( , 2 ) ( 2 , ) and is decreasing on ( 2 , 2 ) .

While some functions are increasing (or decreasing) over their entire domain, many others are not. A value of the input where a function changes from increasing to decreasing (as we go from left to right, that is, as the input variable increases) is called a local maximum    . If a function has more than one, we say it has local maxima. Similarly, a value of the input where a function changes from decreasing to increasing as the input variable increases is called a local minimum    . The plural form is “local minima.” Together, local maxima and minima are called local extrema    , or local extreme values, of the function. (The singular form is “extremum.”) Often, the term local is replaced by the term relative . In this text, we will use the term local .

Clearly, a function is neither increasing nor decreasing on an interval where it is constant. A function is also neither increasing nor decreasing at extrema. Note that we have to speak of local extrema, because any given local extremum as defined here is not necessarily the highest maximum or lowest minimum in the function’s entire domain.

For the function whose graph is shown in [link] , the local maximum is 16, and it occurs at x = −2. The local minimum is −16 and it occurs at x = 2.

To locate the local maxima and minima from a graph, we need to observe the graph to determine where the graph attains its highest and lowest points, respectively, within an open interval. Like the summit of a roller coaster, the graph of a function is higher at a local maximum than at nearby points on both sides. The graph will also be lower at a local minimum than at neighboring points. [link] illustrates these ideas for a local maximum.

Graph of a polynomial that shows the increasing and decreasing intervals and local maximum.
Definition of a local maximum

These observations lead us to a formal definition of local extrema.

Local minima and local maxima

A function f is an increasing function    on an open interval if f ( b ) > f ( a ) for every a , b interval where b > a .

A function f is a decreasing function    on an open interval if f ( b ) < f ( a ) for every a , b interval where b > a .

A function f has a local maximum at a point b in an open interval ( a , c ) if f ( b ) is greater than or equal to f ( x ) for every point x ( x does not equal b ) in the interval. Likewise, f has a local minimum at a point b in ( a , c ) if f ( b ) is less than or equal to f ( x ) for every x ( x does not equal b ) in the interval.

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
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Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
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Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
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"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
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A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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