Algebra 1 · F.IF.B.4 · A.SSE.B.3B

Turning points, maximums and minimums

A graph has several hills. Which one is “the maximum”, and how do you say where it is?

You need

Increasing, decreasing, constant, positive and negative intervalsAlgebra 1 Function notation in a real situationAlgebra 1

Unlocks

Degree, end behavior and the shape of a polynomial graphAlgebra 2 Maximum, minimum and range of a quadratic functionAlgebra 1

Where you are 10 in Features of a graph

Functions13

What a function is5

Graphs of functions5

Features of a graph10

▴ 5 above Increasing, decreasing, constant, positive and negative intervals Average rate of change4 Concavity: when the rate of change itself changes Continuous graphs, jumps and holes

Turning points, maximums and minimums

Why this exists

The elevation profile of a mountain stage in a cycling race has several summits. Ask “where is the top?” and there are two fair answers: the top of each climb, and the top of the whole stage. A rider going over a lower summit has still reached a top, even though a higher one comes later.

Mathematics keeps both answers by giving them different names, relative and absolute. It also insists on two numbers each time: how high (an output) and where (an input). “The highest point of the route was 1{,}840 m, at 62 km” says both; “the top is 62” leaves the reader guessing which.

Definition

Let f be a function and c an input in its domain.

f(c) is a relative maximum (also called a local maximum) if f(c) is at least as large as every output f(x) for inputs x near c, on both sides of c.
f(c) is a relative minimum (or local minimum) if f(c) is no larger than every output for inputs near c, on both sides.
f(c) is the absolute maximum if f(c) \ge f(x) for every x in the domain, and the absolute minimum if f(c) \le f(x) for every x in the domain.
A turning point is a point where the graph switches from rising to falling (a peak) or from falling to rising (a valley), reading left to right.

“Near c” means within some interval around c, however small, that reaches to both sides of it. The plurals are maxima and minima.

A maximum or minimum is an output, a value of f. Where it happens is an input. Always state both: “the relative maximum is 4, at x = -3”, or give the point (-3, 4).

The convention used here. Because a relative maximum or minimum is compared with inputs on both sides, an endpoint of a restricted domain is never called a relative maximum or minimum in this lesson. Some books do count endpoints, so check which convention a book uses. An endpoint can still give the absolute maximum or minimum.

Intuition

Read the graph in Figure 1 as a road, walking left to right.

Figure 1Two peaks and a valley. Only the higher peak is the absolute maximum.

You climb to the point (-3, 4) and then start going down: a peak, so a turning point. Standing there, the road just behind you and just ahead of you is lower, so 4 is a relative maximum. You go down to (-1, 1) and start climbing: a valley, and 1 is a relative minimum. You climb to (2, 5), higher than anything else on the road, and the road falls away for good.

“Relative” means relative to the neighbors. The peak at (-3, 4) only has to beat the outputs near it, and it does. The absolute maximum has to beat every output, and only 5 does that. The rising and falling stretches between these points are the increasing and decreasing intervals; the turning points are where one kind of stretch hands over to the other.

There is no absolute minimum. Both arrows point down, so the graph keeps falling without end on both sides (see End behavior). Name any low output and the graph goes lower.

Results

Every peak gives a relative maximum, and every valley a relative minimum.
A function has at most one absolute maximum value, but that value may be reached at more than one input. The same holds for the absolute minimum.
An absolute maximum at an input that is not an endpoint is also a relative maximum. So the absolute maximum, if there is one, is the largest of the relative maxima and the endpoint values. The same holds for minima.
If an end of the graph rises without end, there is no absolute maximum. If an end falls without end, there is no absolute minimum.
If the domain is a closed interval a \le x \le b and the graph has no breaks, the absolute maximum and minimum both exist.

Why it is true

A peak is a relative maximum. Just left of a peak at c the graph is rising, so those outputs are smaller than f(c). Just right of it the graph is falling, so those outputs are smaller too. So f(c) is at least as large as every output near c. A valley is the same argument turned upside down.

There is only one absolute maximum value. Suppose M and N were both absolute maximum values. M is at least every output, and N is an output, so M \ge N. In the same way N \ge M. So M = N. Nothing stops this one value being reached at two inputs: Graph A in the exercises reaches 4 twice.

An absolute maximum away from the ends is a relative one. If f(c) is at least every output in the whole domain, it is at least every output near c. Since c is not an endpoint, there are inputs on both sides of it. So to find the absolute maximum it is enough to compare the relative maxima with the endpoint values.

A rising end allows no absolute maximum. Whatever output you call the largest, further along that end there are larger outputs.

The last result, that a graph without breaks on a closed interval always has both, is not proved here. It needs a careful meaning of “no breaks”, which comes in later courses.

Worked examples

Example 1

Reading a graph with two peaks

Use Figure 1. List the turning points, the relative maxima and minima, and the absolute maximum and minimum.

Solution

The graph switches direction three times, so the turning points are (-3, 4), (-1, 1) and (2, 5).

The peaks give the relative maxima: 4, at x = -3, and 5, at x = 2. The valley gives the relative minimum: 1, at x = -1.

For the absolute maximum, compare the relative maxima. There are no endpoints, since both arrows say the graph goes on. The larger is 5, so the absolute maximum is 5, at x = 2.

There is no absolute minimum: both ends fall without end, so no output is the smallest. The relative minimum 1 is only the lowest output near x = -1.

Example 2

An endpoint wins

The function g has domain 0 \le x \le 8, graphed in Figure 2. Find its relative and absolute maxima and minima.

Figure 2The right endpoint is higher than the peak, so it gives the absolute maximum.

Solution

The peak at (2, 5) gives a relative maximum 5, at x = 2. The valley at (5, -1) gives a relative minimum -1, at x = 5. The endpoints (0, 2) and (8, 6) are not relative maxima or minima under our convention, since each has inputs on one side only.

For the absolute values, compare the turning points with the endpoint values g(0) = 2 and g(8) = 6.

Largest output: the candidates are 5, 2 and 6. The absolute maximum is 6, at x = 8. That point is not a turning point: the graph is still rising when the domain stops.

Smallest output: the candidates are -1, 2 and 6. The absolute minimum is -1, at x = 5, which is also the relative minimum.

Example 3

The highest point of a mountain stage

Figure 3 is the elevation profile of a made-up cycling stage. The elevation is h(d) meters after d kilometers. Between marked points the road climbs or descends without a break in direction.

Figure 3A made-up mountain stage. The table in Example 3 gives the marked points.

Point	Distance d (km)	Elevation h(d) (m)
A (start)	0	420
B	38	1{,}250
C	51	760
D	62	1{,}840
E	95	610
F (finish)	120	1{,}120

Give the relative and absolute maxima and minima of h, in sentences with units. Then find the height gained on the climb from C to D.

Solution

The summits B and D are peaks, so the relative maxima are h(38) = 1{,}250 m and h(62) = 1{,}840 m. The valleys C and E give the relative minima, h(51) = 760 m and h(95) = 610 m. A and F are endpoints, so they are not counted as relative extremes.

Absolute maximum: compare 1{,}250, 1{,}840 and the endpoint values 420 and 1{,}120. The highest point of the route was 1{,}840 m, at 62 km.

Absolute minimum: compare 760, 610, 420 and 1{,}120. The lowest point of the route was 420 m, at the start (0 km). It is an endpoint, not a valley: the lowest valley, E, is 190 m higher.

From C to D the road gains 1{,}840 - 760 = 1{,}080 m over 62 - 51 = 11 km.

Example 4

A variable star, from a table

A variable star is a star whose brightness rises and falls. An observer measures one every day at midnight. The numbers are made up, in made-up brightness units.

Day	Brightness	Day	Brightness
0	52	6	42
1	61	7	44
2	66	8	51
3	63	9	60
4	55	10	65
5	47

Estimate the star’s relative maximum and minimum brightness and when they happen.

Solution

The reading on day 2, 66, is larger than the readings on days 1 and 3, so the star brightened and then dimmed again between day 1 and day 3. Its peak lies somewhere in that span, and it is at least 66, since 66 was measured. The best the table can say: a relative maximum of about 66 units, near day 2.

The reading on day 6, 42, is smaller than those on days 5 and 7. So there is a relative minimum of about 42 units, near day 6: at most 42, somewhere between day 5 and day 7.

These are estimates because a table says nothing about the brightness between midnights. For all the table shows, the true peak could be 68 units at noon on day 2. The table also stops while the star is brightening again (60, then 65), so it cannot say how bright the next peak will be.

Example 5

A valley above a peak

Figure 4 shows a function k defined for every real number. The left end falls without end and the right end rises without end. Find the relative and absolute maxima and minima.

Figure 4The valley at height 4 is higher than the peak at height 2.

Solution

Peaks at (-4, 2) and (1, 6): relative maxima 2, at x = -4, and 6, at x = 1. Valleys at (-2, -1) and (3, 4): relative minima -1, at x = -2, and 4, at x = 3.

The relative minimum 4 is larger than the relative maximum 2. Nothing is wrong: each is compared only with its own neighbors.

There is no absolute maximum, because the right end rises without end, and no absolute minimum, because the left end falls without end. A function can have relative extremes and no absolute ones.

Example 6

Finding the extremes from a rule

Find the absolute maximum and minimum of

f(x) = 4 - (x - 1)^2, \qquad -1 \le x \le 4.

Solution

A square is never negative, so (x - 1)^2 \ge 0 and f(x) \le 4 for every x. Equality needs (x - 1)^2 = 0, that is x = 1, which is in the domain. So the absolute maximum is f(1) = 4, at x = 1. The graph is a parabola and (1, 4) is its vertex, its only turning point.

The further x is from 1, the larger (x - 1)^2 and the smaller f(x). The input in the domain furthest from 1 is the endpoint x = 4, three units away; the other endpoint, x = -1, is only two away. So the absolute minimum is at an endpoint:

f(4) = 4 - 3^2 = -5, \qquad f(-1) = 4 - 2^2 = 0.

The absolute minimum is -5, at x = 4. There is no relative minimum: the graph has no valley.

Boundaries

Endpoints. Under the convention here, an endpoint is never a relative maximum or minimum, but it can give an absolute one (Examples 2 and 3). A book that counts endpoints will list more relative extremes for the same graph.
Flat stretches. On a level piece of graph, every point is at least as large as its neighbors and no larger than them, so by the definitions each is both a relative maximum and a relative minimum, although no turning point is there.
Breaks and open ends. On 0 \le x < 2, the function f(x) = x has outputs as close to 2 as you like but never 2, so it has no absolute maximum even though its domain is short. The closed-interval result needs both endpoints included and no breaks.
Reading is estimating. From a graph without marked coordinates, or from a table, a maximum or minimum and its input are estimates.

Where this goes next

For a parabola the single turning point is the vertex, and Maximum, minimum and range of a quadratic function uses it to find the largest or smallest output and the range. Maximum, minimum and projectile problems then finds the best possible value in a real situation, such as the height of a thrown ball, by algebra. Degree, end behavior and the shape of a polynomial graph limits how many turning points a polynomial graph can have. In calculus, Maxima, minima and critical points locates turning points exactly from a rule, and Optimization is the “compare turning points with the endpoints” idea of Example 2, made precise.

Exercises

Reading graphs

Use Figure 5.

Figure 5Graphs A and B for Exercises 1 to 4.

For Graph A, list each turning point and say whether it gives a relative maximum or a relative minimum.
For Graph A, give the absolute maximum and absolute minimum, if they exist, and where each happens.
Graph B has domain -4 \le x \le 4. Give its relative maxima and minima.
Give the absolute maximum and absolute minimum of Graph B, and where each happens. Which of these happen at a point that is not a turning point?

Reading tables

A function p whose graph has no breaks has p(0) = 3, p(1) = 7, p(2) = 8, p(3) = 6, p(4) = 2, p(5) = 1 and p(6) = 4. Estimate its relative maximum and relative minimum and where each happens.
A second made-up variable star gave these readings on days 0 to 8: 30, 24, 19, 21, 28, 35, 37, 33, 26. Estimate its brightest and dimmest values in the period and when they happened.

From a rule

f(x) = |x - 2| + 1, for every real x. Find any turning point and the absolute maximum and minimum, if they exist.
g(x) = 2x - 1, for 0 \le x \le 5. Find any turning points and the absolute maximum and minimum.
q(x) = x^2, for -2 \le x \le 3. Find the absolute maximum and minimum.

In context

Another made-up stage has these marked points, as (distance in km, elevation in m), with the road changing direction only at marked points: (0, 230), (24, 980), (40, 540), (71, 1{,}620), (88, 1{,}150), (104, 1{,}410), (130, 300). Give its relative maxima and the absolute maximum and minimum, in sentences with units.
Genevieve rides the stage in Exercise 10 and says the lowest point was 540 m, at 40 km. Is that the absolute minimum? Explain.
For a variable star, B(t) is the brightness in units after t days. B(4) = 58 is a relative maximum and B(9) = 31 is a relative minimum. Say what each means in a sentence.

Explain

Can a function have two different absolute maximum values? Can it reach its absolute maximum at two different inputs? Explain.
Describe a function defined for every real number that has a relative maximum but no absolute maximum.
Let c(x) = 3 for every real x. Which outputs are relative maxima, relative minima, the absolute maximum and the absolute minimum? Does the graph have a turning point?
In Exercise 5, could p reach the value 9 somewhere between x = 1 and x = 3? Could p(2) be 9? Explain.

Answers

(-3, 4) and (3, 4) are peaks, giving relative maxima 4 at x = -3 and at x = 3. (0, -2) is a valley, giving a relative minimum -2 at x = 0.
The absolute maximum is 4, reached at both x = -3 and x = 3. There is no absolute minimum: both ends fall without end.
Relative maximum 3, at x = -1; relative minimum 0, at x = 2. The endpoints are not counted.
Absolute maximum 3, at x = -1 and at x = 4; absolute minimum -3, at x = -4. The candidates are 3, 0 and the endpoint values -3 and 3. The minimum at x = -4 and the maximum at the endpoint x = 4 are not at turning points.
Relative maximum about 8, near x = 2: at least 8, somewhere between x = 1 and x = 3, since p(1) = 7 and p(3) = 6 are smaller. Relative minimum about 1, near x = 5: at most 1, between x = 4 and x = 6.
Brightest about 37 units, near day 6 (between days 5 and 7); dimmest about 19 units, near day 2 (between days 1 and 3). Every other reading is between 19 and 37.
Turning point (2, 1), a corner where the graph switches from falling to rising. The absolute minimum is 1, at x = 2, since |x - 2| \ge 0. No absolute maximum: both arms rise without end.
No turning points: the graph rises all the way. The absolute minimum is g(0) = -1, at x = 0, and the absolute maximum is g(5) = 9, at x = 5, both at endpoints.
Absolute minimum q(0) = 0, at x = 0, a turning point. Absolute maximum q(3) = 9, at x = 3, since q(-2) = 4 is smaller.
The relative maxima are 980 m at 24 km, 1{,}620 m at 71 km and 1{,}410 m at 104 km. The highest point was 1{,}620 m, at 71 km. The lowest point was 230 m, at the start (0 km); the finish, at 300 m, is higher.
No. 540 m is a relative minimum, the lowest point near 40 km. The start, at 230 m, is lower, and so is the finish, at 300 m. An endpoint value must be compared too.
On day 4 the star reached a peak brightness of 58 units: at no time shortly before or after was it brighter. On day 9 it fell to a low of 31 units: at no time shortly before or after was it dimmer.
No to the first: if M and N were both absolute maximum values, then M \ge N and N \ge M, so M = N. Yes to the second: Graph A reaches 4 at x = -3 and x = 3.
Any graph with a peak whose end rises without end, for example Figure 4: it has a relative maximum 6 at x = 1, but its right end rises past every output.
Every input gives the output 3, which is at least, and at most, every nearby output. So by the definitions here every output is a relative maximum and a relative minimum, and 3 is both the absolute maximum and the absolute minimum. There is no turning point: the graph never rises or falls. Some books use a strict definition (larger, or smaller, than every other nearby output); under it c has no relative maxima or minima.
Yes to the first: Exercise 5 gives no values between x = 1 and x = 3 other than p(2), so the graph could rise to 9 there and come back. No to the second: we are told p(2) = 8.