Applying Derivatives to Graphs - Maximums and Minimums

In this unit, we will start to apply what we learned about derivatives into graphs. We will see our theoretical knowledge being applied to graphs and this will start to show us how derivatives can be applied to the real world.

The first part of our application of derivatives to graphs will involve the concept of Maximums and Minimums. As we may be aware, we can plot graphs of functions that represent a large amount of information. For example, profits of a company, population of a species, the speed of a car etc. Maximum and Minimums (of a graph of a function) is the method of identifying where values of a function are at their Maximum or Minimum.

For example, say we have the graph of a function that plots the amount of rainfall per month, graph showing rainfall amounts by month with peak in january

We have a point where the value of rainfall is at its greatest, at some point in January or February, this is the Maximum. We have a point where the value of rainfall is at its lowest, at some point in June or July, this is the Minimum.

Defining Maximums and Minimums

Below we will define different types of Maximums and Minimums, try to not be put off by the introduction of new terms. It may at first seem strange but once we start with the examples it will become more obvious what they are.

Definition - Maximum / Global Maximum, and Maximum Point

Maximum / Global Maximum - A function of x, f(x), has the maximum at the point a, in that function, as long as f(a) is greater than all other values of x in f(x). This is also classed as a Global Extreme.

Maximum Point - The maximum point is the point (a, f(a)) where f(a) is greater than all other possible values along the function f(x)

Definition - Local and Relative Maximum

Local / Relative Maximum - A function f(x) has a local or relative maximum at the value a, if f(a) is greater than the values of f(x) that are “near” it. This definition is slightly less strict but will become more obvious once we start to look at examples on graphs. This is also classed as a Local Extreme.

We apply the same definition for Minimums, swapping “maximum” for minimum.

Definition - Minimum / Global Minimum , and Minimum Point

Minimum / Global Minimum - A function of x, f(x), has the Minimum at the point a, in that function, as long as f(a) is lower than all other values of x in f(x). This is also classed as a Global Extreme.

Minimum Point - The Minimum point is the point (a, f(a)) where f(a) is lower than all other possible values along the function f(x).

Definition - Local and Relative Minimum

Local / Relative Maximum - A function f(x) has a local or relative maximum at the value a, if f(a) is greater than the values of f(x) that are “near” it. This definition is slightly less strict but will become more obvious once we start to look at examples on graphs. This is also classed as a Local Extreme.

Definition - Global Extreme and Local Extreme

Global Extreme - Either a Global Maximum or a Global Minimum

Local Extreme - Either a Local Maximum or a Local Minimum

Have a look at the graph of function f(x) below that incorporates Global and Local Maximums and Minimums. x-y graph showing function f(x) with Global Maximum, Global Minimum, Local Maximum, and Local Minimum points with coordinate values.

So for this graph, just by looking at the curve, we can see the greatest or highest value on it is at point (a, f(a)) this is the Global Maximum. We can also see the lowest value is at point (b, f(b)), this is the Global Minimum.

Also, we can see we have two other points in the curve that have their own maximum and minimum values when just considering the values around it that are “near”. This is the Local Maximum at (a1, f(a1)) and the Local Minimum at (b1, f(b1)).

Determining Maximum and Minimums of a Function

Now, instead of just looking at a graph to find its maximum and minimums, we will learn how to use Mathematics to find them.

When a point is not a Maximum or Minimum

For Maximum and Minimum values - one characteristic they have is that they are “flat” on the graph. That is, the point where they are a Maximum or Minimum, the tangent to that point is 0. We can use differentiation to determine what the gradient of the tangent at that point is.

So, say we are looking at a point a, in a function f(x) and we want to know if it's a maximum or minimum, if we differentiate that function and input a, f’(a) will tell us if the gradient on the tangent is equal to 0. If it is not equal to zero, then it is not a maximum or minimum.

Mathematically, we write this like:

For function f(x), the function is not a maximum or minimum at point a, if,

f’(a) > 0 or f’(a) < 0

The other two possibilities for the value of f’(a) is f’(a) = 0 or f’(a) is undefined.

Definition - Local Extreme - with derivative

A Function is a Local Extreme (either local minimum or local maximum) at a point a if,

f’(a) = 0 or f’(a) is undefined

Recalling that a Function is a Local Extreme (either local minimum or local maximum) at a point a if,

f’(a) = 0 or f’(a) is undefined

Work through the following examples, first calculate the derivative, then figure out when the derivative is equal to zero or is undefined, this will give you the values for the local extremes.

The steps for this are,

1) Differentiate the function f(x) to get f’(x)

2) Equate f’(x) to zero. f’(x) = 0 or f’(x) is undefined

3) Solve for x in f’(x) = 0

Note, we won't be looking for a local maximum or minimum just yet, finding the Local Extreme will tell us we have either a local maximum or a local minimum.

These values we get are possible Local Extremes, the other term for them is Critical Numbers.

Local Extremes

Example

Find the value/ values for the possible local extremes of the following functions:

a) f(x) = x2 - 4x

So we are looking for the possible local extremes of the function f(x). Recall that possible local extremes occur when:

f '(x) = 0 or when undefined.

So the first step is to find the derivative of f(x) = x2 - 4x

This is: f '(x) = 2x - 4

Next we look for the x-values when f '(x) = 0 to give us the values for the possible local extreme.

This is: f '(x) = 0 → possible local extreme

This is: f '(x) = 2x - 4 = 0 → possible local extreme

So we solve for: 2x - 4 = 0 to give us the value for the possible local extreme.

2x - 4 = 0

2x = 4

x = 2

We have a possible local extreme at x = 2


b) f(x) = 3x2 + 12x - 1

So we know that the possible local extreme happens at the value/ values of x that give f '(x) = 0

Find f '(x) then equate to zero and solve.

f(x) = 3x2 + 12x - 1

f '(x) = (2)3x2-1 + 12(1) - 0

f '(x) = 6x + 12

Next equate f '(x)= 0 to find the x-value/ values for the possible local extreme of the function.

6x + 12 = 0

6x = -12

x = -2


c) f(x) = x3 - 27x

First find f '(x)

f(x) = x3 - 27x

f '(x) = 3x2 - 27

Next equate f '(x)= 0 to find the value/ values for the possible local extremes.

3x2 - 27 = 0

Solve for x to find the value/ values for the possible local extremes.

3x2 - 27 = 0

3x2 = 27

x2 = 9

x = √9

x = 3 or - 3 [(-3)2 = 9!]

Here we have two values for our possible local extreme,   x = 3 and x = -3

Note: we have identified possible local extremes (critical numbers). What we have done is ruled out, that apart from our answer/answers, there are no other possible local extremes. We can identify local extremes by either looking at a graph or through a method we will teach in the near future.

Example 1 - Local Extremes

Find the values of the points (x, f(x)) of the possible local extremes (PLE) for the following polynomials.

a) f(x) = 2x3 - 21x2 + 72x + 7

So, to find the possible local extreme (PLE) values, find f '(x) and equate it to zero. The local extreme occurs at the values of x, where f '(x) = 0

f(x) = 2x3 - 21x2 + 72x + 7

f '(x) = 6x2 - 42x + 72 + 0

→Look for a common factor →

= 6(x2 - 7x + 12)

→Factorize →

= 6(x - 3)(x - 4)

Now equate f '(x) = 0

f(x) = 6(x - 3)(x - 4) = 0

f '(x) = 0 when either x = 3 or x = 4

Now the last part; write these x-values as points (x, y)

x = 3 → (3, f(3))

f(3) = 2(3)3 - 21(3)2 + 72(3) + 7

= 54 - 189 + 216 + 7

= 88

→ (3, 88)

x = 4 → (4, f(4))

f(4) = 2(4)3 - 21(4)2 + 72(4) + 7

= 128 - 336 + 288 + 7

= 87

→ (4, 87)


b) f(x) = 2x3 - 3x2 - 12x + 6

Finding possible local extremes (PLE) values, we need f '(x) = 0 and to find the values of x that satisfy that.

First, find f '(x)

f(x) = 2x3 - 3x2 - 12x + 6

f '(x) = 6x2 - 6x - 12 + 0

→ common factor →

= 6(x2 - x - 2)

→ "What multiplies to -2 and adds to -1? → 1 & -2

f '(x) = 6(x + 1)(x - 2)

Now solve for f '(x) = 0 to give us the values for the possible local extremes.

f '(x) = 6(x + 1)(x - 2)

→ f '(x) = 0 when x = -1 & x = -2

Lastly, find the y-values for these x-points

x = -1 → (-1, f(-1))

f(-1) = 2(-1)3 - 3(-1)2 - 12(-1) + 6

= 2(-1) - 3(1) + 12 + 6

= -2 - 3 + 12 + 6

= 13

→ (-1, -13) is a point for one of our possible local extremes.

x = 2 → (2, f(2))

f(2) = 2(2)3 - 3(2)2 - 12(2) + 6

= 16 - 12 - 24 + 6

= -14

→ (2, -14) is a point for one of our possible local extremes.

Often, when we look for possible Local Extremes (Critical Numbers), it is where the values where the derivative of the function is equal to zero. However, it is important to remember that the possible Local Extremes can occur for the values of x that make the derivative of the function undefined. For example, if we had,

f’(x) = 1/x , a value that would make f’(x) undefined would be x = 0

As 1/0 is impossible and therefore undefined.

This is a good tip when looking for undefined values of a derivative, any time we see x as a denominator, (like [ 3 / (x2) ] or [ 1 / (x - 1) ] ) then look for the values of x that make the denominator zero. Any time a denominator is zero, the equation will be undefined.

Local Extremes - where f’(x) is undefined

worked example showing how to determine possible local extremes for the function f(x) = x to the power of one half
part 1 of a worked example showing how to determine possible local extremes for the function f(x) = (three-fifths multiplied by x to the power of five thirds) subtract three halves multipled by x to the power of  two-thirds

part 2 of a worked example showing how to determine possible local extremes for the function f(x) = (three-fifths multiplied by x to the power of five thirds) subtract three halves multipled by x to the power of  two-thirds
part 3 of a worked example showing how to determine possible local extremes for the function f(x) = (three-fifths multiplied by x to the power of five thirds) subtract three halves multipled by x to the power of  two-thirds

Endpoint Extremes

We have been working with functions that are defined on an open interval, that is, there are no restrictions on the range of the values of x.

We may deal with functions that have restricted values of x, for example,

f(x) = x2 + 1 for ( -2 ≤ x ≤ 2 )

In that case the endpoints would be x = -2 and x = 2

It may be the case that one or both endpoints of a function are Local Extremes.

Say we have the function f(x), for the values in the interval a ≤ x ≤ bx-y graph of f(x) for the x-values between a and b with local maximum and minimum points annotated.

It may be the case that the endpoints of the function are Local Extremes, it is for that reason, that we must include endpoints in the possible Local Extreme values (Critical Numbers)

Up until now, we have been working out if a function has any possible Local Extremes (Local Extremes are either Local Maximums or Local Minimums - a value that is the largest or smallest for the values that are “near”). That was the method of finding the values of x that make the derivative of the function equal zero or undefined or if it is an endpoint in an interval in a function. These values of x have the possibility of being a Local Extreme, but it is not certain whether or not they are yet.

These values of x that make the derivative of the function either equal to zero or undefined or if it is an endpoint in an interval in a function, they are called Critical Numbers.

When we learn how to confirm if these values (or Critical Numbers) are Local Extremes, we will then call these Critical Points.

Definition - Critical Number and Critical Point

Critical Number - Possible Local Extreme, the values in a function that make the derivative of that function equal to zero or undefined or is an endpoint

The value, a, in the function f(x), that makes f’(a) = 0 or undefined or is an endpoint.

Critical Point - A confirmed Local Extreme, the values in a function that make the derivative of that function equal to zero or undefined or is an endpoint. Also, these numbers must be confirmed as Local Extremes.

The point (a, f(a)), that satisfies f’(a) = 0 or undefined as well as it must be a confirmed point that is a Local Extreme.