WEBVTT

6fa78cf7-b1d7-409d-a3fe-f17f5083d624-0
00:00:05.520 --> 00:00:06.440
Hi, I'm Fiona.

a86fff29-070a-4f8e-81ff-2ba1912986e2-0
00:00:06.720 --> 00:00:10.880
Let's have a look at Timura 2021,
paper 2 and question 9.

3cccd736-961b-4458-a78a-8133d0219e19-0
00:00:11.280 --> 00:00:15.400
We're asked to consider the following
statements about a polynomial F of X.

4560295c-0f1a-4822-8367-5219a3dcc7be-0
00:00:15.720 --> 00:00:22.051
Statement 1F of X equals PX cubed plus QX
squared plus RX plus S where P is not

4560295c-0f1a-4822-8367-5219a3dcc7be-1
00:00:22.051 --> 00:00:23.160
equal to zero.

a685b5a8-79b9-4181-ab31-c9a05c675f01-0
00:00:23.480 --> 00:00:30.040
Statement 2 says there is a real number T
for which P-T equals 0.

bfda3ccc-ef2d-4b24-9f98-1debf12737c5-0
00:00:30.040 --> 00:00:33.680
Let me just make that dash bit more
noticeable.

569dc63e-846f-47b0-a72c-92a892c65135-0
00:00:34.680 --> 00:00:38.373
And statement three,
there are real numbers U&

569dc63e-846f-47b0-a72c-92a892c65135-1
00:00:38.373 --> 00:00:41.560
V for which F of U * F of V is less than
0.

06cce756-dfa9-4f16-a2b8-2e1568feae67-0
00:00:41.920 --> 00:00:47.483
We're asked which of these statements is
or are sufficient for the equation F of X

06cce756-dfa9-4f16-a2b8-2e1568feae67-1
00:00:47.483 --> 00:00:49.360
= 0 to have a real solution.

81482242-2119-4542-b8f2-3a8fa721111a-0
00:00:50.160 --> 00:00:52.066
Well,
the first thing I noticed with this

81482242-2119-4542-b8f2-3a8fa721111a-1
00:00:52.066 --> 00:00:54.880
question is that it's always helpful to
label our statements.

42bbede3-9bcd-4422-b9ad-0d81ac480e16-0
00:00:55.080 --> 00:00:59.172
Statements 1-2 and three have a label
given in this question,

42bbede3-9bcd-4422-b9ad-0d81ac480e16-1
00:00:59.172 --> 00:01:01.680
but this statement at the end doesn't.

8fe0186c-7887-4d55-9ba1-b75d44c2d575-0
00:01:01.680 --> 00:01:05.600
So I'm going to let P be the statement.

3a239a07-f6ff-423d-ab08-96adb3ef7a7d-0
00:01:05.920 --> 00:01:10.209
The equation F of X = 0 has a real
solution,

3a239a07-f6ff-423d-ab08-96adb3ef7a7d-1
00:01:10.209 --> 00:01:16.120
and this is just the existence of at
least one real solution.

974db389-507e-4ee9-8d19-c8616a6d88c6-0
00:01:16.240 --> 00:01:18.360
It doesn't have to be 1,
it could be more than one.

1450d8d3-e488-4819-99a0-1cb3b523b44c-0
00:01:18.600 --> 00:01:22.532
But we when we think about statement P
here,

1450d8d3-e488-4819-99a0-1cb3b523b44c-1
00:01:22.532 --> 00:01:29.699
what we're saying is that our function F
of X needs to pass through or meet the X

1450d8d3-e488-4819-99a0-1cb3b523b44c-2
00:01:29.699 --> 00:01:31.359
axis at least once.

81dbd049-6877-45e5-b525-6f63f3c37c25-0
00:01:37.240 --> 00:01:40.812
OK,
so now let's go through statement 1-2 and

81dbd049-6877-45e5-b525-6f63f3c37c25-1
00:01:40.812 --> 00:01:46.714
three and think about whether each one of
them is or are sufficient for our

81dbd049-6877-45e5-b525-6f63f3c37c25-2
00:01:46.714 --> 00:01:49.200
statement that we've labelled P.

a4f311aa-e866-4571-941f-b7a8307d1e59-0
00:01:49.680 --> 00:01:53.432
Now, just a little aside,
when we're looking at whether one

a4f311aa-e866-4571-941f-b7a8307d1e59-1
00:01:53.432 --> 00:01:56.372
statement is sufficient for another
statement,

a4f311aa-e866-4571-941f-b7a8307d1e59-2
00:01:56.372 --> 00:02:00.000
it's helpful to think about the direction
of implication.

0a00d841-898e-4544-9431-a31c5506cc22-0
00:02:00.280 --> 00:02:04.530
And so if I start with statement 1,
and I'm thinking is statement 1

0a00d841-898e-4544-9431-a31c5506cc22-1
00:02:04.530 --> 00:02:08.155
sufficient for P,
then mathematically we would write this

0a00d841-898e-4544-9431-a31c5506cc22-2
00:02:08.155 --> 00:02:09.280
double arrow here.

0176392d-6cdd-44a9-84d5-68b698b717d7-0
00:02:09.280 --> 00:02:14.040
So statement one is sufficient for P if
this holds.

af7c7053-521a-4039-9acf-56d111d9e7e9-0
00:02:15.160 --> 00:02:19.335
And it's helpful to think about it this
way because it means we start with

af7c7053-521a-4039-9acf-56d111d9e7e9-1
00:02:19.335 --> 00:02:21.840
statement one and considering statement
one.

d9b17e81-7868-4ea1-81dc-3409afcee199-0
00:02:22.200 --> 00:02:28.533
And we think is there a way in which we
can investigate whether statement one

d9b17e81-7868-4ea1-81dc-3409afcee199-1
00:02:28.533 --> 00:02:33.000
guarantees P or whether P follows on from
statement 1.

ce57406a-ff3a-42bb-af84-2009f6fe2f58-0
00:02:33.240 --> 00:02:42.335
So that is the same as checking whether
statement one is sufficient for P OK,

ce57406a-ff3a-42bb-af84-2009f6fe2f58-1
00:02:42.335 --> 00:02:45.600
what does statement one say?

89ec2658-f874-4902-b2fd-546f70476a8f-0
00:02:45.800 --> 00:02:51.716
Statement one gives us a polynomial PX
cubed plus QX squared plus RX plus S But

89ec2658-f874-4902-b2fd-546f70476a8f-1
00:02:51.716 --> 00:02:57.560
the key part of the information in this
statement is that P is not equal to 0.

58b21fdc-2d07-4d13-9222-ee30d7bc3d72-0
00:02:57.760 --> 00:03:01.419
And if P is not equal to 0P being the
coefficient of X ^3,

58b21fdc-2d07-4d13-9222-ee30d7bc3d72-1
00:03:01.419 --> 00:03:04.520
then that means that we are dealing with
a cubic.

d89d544b-4eb3-43aa-89fd-a4bf629b574e-0
00:03:06.080 --> 00:03:10.734
So in statement one,
it's just telling us that we have to have

d89d544b-4eb3-43aa-89fd-a4bf629b574e-1
00:03:10.734 --> 00:03:13.320
a cubic as our polynomial function.

1d1772f4-7bde-4bf6-9f05-124e9af4a9de-0
00:03:13.960 --> 00:03:15.760
Now let's think about cubics.

c91689c0-bb09-4aed-a5d5-3812f1407c23-0
00:03:16.200 --> 00:03:21.717
If we have a coefficient of X ^3 in our
cubic that is greater than 0,

c91689c0-bb09-4aed-a5d5-3812f1407c23-1
00:03:21.717 --> 00:03:27.157
then we would get a function or a cubic
that looked a bit like this,

c91689c0-bb09-4aed-a5d5-3812f1407c23-2
00:03:27.157 --> 00:03:33.384
and depending on the coefficients would
depend where our stationary points are

c91689c0-bb09-4aed-a5d5-3812f1407c23-3
00:03:33.384 --> 00:03:35.040
and things like this.

23048a61-372e-45a5-8396-c2bce9246bb8-0
00:03:35.320 --> 00:03:40.260
But in general,
all cubics for which P is greater than 0

23048a61-372e-45a5-8396-c2bce9246bb8-1
00:03:40.260 --> 00:03:45.200
start with values very large and negative,
so down here.

0c2912d9-0f3f-459f-a447-3866ba6142d9-0
00:03:45.520 --> 00:03:53.053
And then as X increases they end up
getting very large and positive and so

0c2912d9-0f3f-459f-a447-3866ba6142d9-1
00:03:53.053 --> 00:03:59.080
somewhere along the way they would have
to pass the X axis.

4db639ef-8aed-4607-b4a1-c6ba5d7b2641-0
00:03:59.440 --> 00:04:06.310
So if I wrote my if I chose my cubic such
that it passed the X axis like this,

4db639ef-8aed-4607-b4a1-c6ba5d7b2641-1
00:04:06.310 --> 00:04:10.920
then that means FX equals 0 would have
one solution.

d88e1843-809d-4025-9560-189586b86557-0
00:04:11.400 --> 00:04:17.100
If my cubic passed the X axis with the X
axis being around here,

d88e1843-809d-4025-9560-189586b86557-1
00:04:17.100 --> 00:04:20.960
then F of X = 0 would have three
solutions.

9f3ecd1a-0056-49a6-93ef-203425395ba7-0
00:04:20.960 --> 00:04:26.114
And there are other scenarios,
but what we can see here is that there

9f3ecd1a-0056-49a6-93ef-203425395ba7-1
00:04:26.114 --> 00:04:30.680
would definitely be at least one real
solution to F of X = 0.

deeb6f09-b2a7-4737-afd3-f5c5f94a05f4-0
00:04:31.400 --> 00:04:35.040
Now let's think about when P is less than
0.

0b1338e3-a6bc-4f84-9365-8f756e5edeb6-0
00:04:35.040 --> 00:04:38.102
Well,
when the coefficient of X ^3 in a cubic

0b1338e3-a6bc-4f84-9365-8f756e5edeb6-1
00:04:38.102 --> 00:04:41.231
equation or in a cubic polynomial is
negative,

0b1338e3-a6bc-4f84-9365-8f756e5edeb6-2
00:04:41.231 --> 00:04:45.160
then that just means that we would get
this sort of shape.

1f31f1fe-2fae-495d-b002-9f1b49e8145b-0
00:04:45.360 --> 00:04:49.827
Now again, depending on the coefficients,
the values of those coefficients,

1f31f1fe-2fae-495d-b002-9f1b49e8145b-1
00:04:49.827 --> 00:04:53.120
we would have stationary points in
different positions.

e7a4509a-9ec5-4067-9852-4f87be7e1b02-0
00:04:53.120 --> 00:04:54.720
We might even have a point Inflexion.

d82c15ca-1f52-4a83-9693-bde65e6f339c-0
00:04:55.760 --> 00:05:01.858
But what we want to focus on here is that
for values of X for the domain of the

d82c15ca-1f52-4a83-9693-bde65e6f339c-1
00:05:01.858 --> 00:05:05.365
function,
when values of X are very large and

d82c15ca-1f52-4a83-9693-bde65e6f339c-2
00:05:05.365 --> 00:05:09.100
negative,
this function will have very large and

d82c15ca-1f52-4a83-9693-bde65e6f339c-3
00:05:09.100 --> 00:05:10.320
positive values.

54bc2dd0-c79f-4a53-8e24-95d8a975c8f2-0
00:05:10.680 --> 00:05:17.259
And as we increase through the domain,
the function would become very large and

54bc2dd0-c79f-4a53-8e24-95d8a975c8f2-1
00:05:17.259 --> 00:05:18.000
negative.

20d9c5b5-003e-468d-8d54-0ddc90afc652-0
00:05:18.440 --> 00:05:24.023
And all that to say that again,
this cubic has to pass through the X axis

20d9c5b5-003e-468d-8d54-0ddc90afc652-1
00:05:24.023 --> 00:05:25.080
at least once.

b85fee75-1814-4a99-8bed-b360f4a0a30d-0
00:05:25.400 --> 00:05:29.205
So again,
I could have a cubic equation such that

b85fee75-1814-4a99-8bed-b360f4a0a30d-1
00:05:29.205 --> 00:05:33.164
the X axis,
it passes through the X axis like this,

b85fee75-1814-4a99-8bed-b360f4a0a30d-2
00:05:33.164 --> 00:05:37.046
or I could have one where there's a
repeated root,

b85fee75-1814-4a99-8bed-b360f4a0a30d-3
00:05:37.046 --> 00:05:42.754
in which case this does guarantee that F
of X = 0 has to have at least one

b85fee75-1814-4a99-8bed-b360f4a0a30d-4
00:05:42.754 --> 00:05:43.440
solution.

a8e55f1e-3610-4a5c-b9e7-1981090d1af5-0
00:05:43.720 --> 00:05:48.280
That means that statement one is
sufficient for P.

f2ca8e97-bd48-439f-8514-ce43474dcb6a-0
00:05:48.560 --> 00:05:52.873
So let me give it a tick because it is
sufficient for P I'm going to clear some

f2ca8e97-bd48-439f-8514-ce43474dcb6a-1
00:05:52.873 --> 00:05:56.000
space on the board and then we'll move on
to statement 2.

0f8ac22a-80cb-4519-96a4-ed91b93bc232-0
00:06:00.360 --> 00:06:04.010
OK,
now let's look at statement 2 and whether

0f8ac22a-80cb-4519-96a4-ed91b93bc232-1
00:06:04.010 --> 00:06:10.120
statement 2 is sufficient for statement P
Well, what does statement two say?

1ac3887d-0b89-4984-92a0-54903158539b-0
00:06:10.240 --> 00:06:15.840
Statement 2 says that there is a real
number T for which F dash of t = 0.

f22f7df2-efcf-4192-b7b4-60c74bf5bf02-0
00:06:16.200 --> 00:06:22.087
That's telling me that whatever
polynomial I'm dealing with in statement

f22f7df2-efcf-4192-b7b4-60c74bf5bf02-1
00:06:22.087 --> 00:06:24.990
2,
which doesn't have to be a cubic

f22f7df2-efcf-4192-b7b4-60c74bf5bf02-2
00:06:24.990 --> 00:06:26.120
statement one.

f59d43fd-b568-41be-8fdc-3e8311c5f8db-0
00:06:26.320 --> 00:06:28.680
In statement one,
our polynomial had to be a cubic.

87747f2a-3722-42bf-8017-2e291f48fcca-0
00:06:28.960 --> 00:06:32.770
But for statement 2,
we're now having to consider all possible

87747f2a-3722-42bf-8017-2e291f48fcca-1
00:06:32.770 --> 00:06:34.040
polynomial functions.

ba32c895-4a35-46b3-b00d-b201272d4e2a-0
00:06:34.360 --> 00:06:40.099
But the statement 2 is telling us that
for these polynomial functions,

ba32c895-4a35-46b3-b00d-b201272d4e2a-1
00:06:40.099 --> 00:06:45.920
there has to be at least a one real
number T for which F dash of t = 0.

bc1ae30b-6472-44f9-a5d1-9a70d86e0bc5-0
00:06:46.200 --> 00:06:50.400
That's telling us that we have to have at
least one stationary point.

c93d330d-bd86-47df-9bdd-d3c6d2fd8659-0
00:06:55.080 --> 00:06:57.336
OK,
so let's think about a function that has

c93d330d-bd86-47df-9bdd-d3c6d2fd8659-1
00:06:57.336 --> 00:06:58.840
at least one stationary point.

11aa1dc2-9e1b-4f27-b28e-2e0c3e220ec3-0
00:06:59.120 --> 00:07:01.520
I'm going to choose an example that's
helpful.

43224d51-574d-420e-ba13-15baf1ee3639-0
00:07:01.800 --> 00:07:06.400
And for us, when we're considering,
is statement 2 sufficient for P?

af26af85-df3c-4286-86c7-9df54e527199-0
00:07:06.640 --> 00:07:12.828
So I'm going to think about a quadratic,
A quadratic with a negative coefficient

af26af85-df3c-4286-86c7-9df54e527199-1
00:07:12.828 --> 00:07:13.440
of X ^2.

195ba4ba-32ea-4939-831d-ab5913ae14a6-0
00:07:13.440 --> 00:07:14.960
So it's got this N shape.

a6e5e460-3e43-4526-b428-48f3d1356015-0
00:07:15.320 --> 00:07:22.207
And suppose that I had a quadratic
function like this such that the X axis

a6e5e460-3e43-4526-b428-48f3d1356015-1
00:07:22.207 --> 00:07:28.361
was up here and this function,
If we say that let's say T is here,

a6e5e460-3e43-4526-b428-48f3d1356015-2
00:07:28.361 --> 00:07:34.973
then that means that this coordinate
point would be the maximum of this

a6e5e460-3e43-4526-b428-48f3d1356015-3
00:07:34.973 --> 00:07:35.800
function.

7f864114-7831-4d75-8a13-633d318aa0a2-0
00:07:36.120 --> 00:07:40.680
It would it would have coordinate point
TF of TF.

3ee35ceb-b700-4309-b64a-b4c5f8879048-0
00:07:41.520 --> 00:07:46.000
Dash of T would be equal to 0 because
this is a stationary point.

c85fe9e7-7d72-4e18-9a9b-68332255a701-0
00:07:46.920 --> 00:07:52.281
And so this polynomial function does
satisfy statement 2,

c85fe9e7-7d72-4e18-9a9b-68332255a701-1
00:07:52.281 --> 00:07:55.240
but does it satisfy statement P?

bc14f12d-1757-49f8-9027-544f4a9b5303-0
00:07:56.240 --> 00:08:02.320
We need for statement P that F of X = 0
has at least one real solution.

6ac3759d-9304-46ef-aca4-c2d464db3799-0
00:08:02.600 --> 00:08:08.400
But this quadratic doesn't pass through
or meet the X axis at all.

23bf821d-0498-4ff1-a5ce-38f47523dda1-0
00:08:08.600 --> 00:08:14.300
So this quadratic satisfies statement 2,
but does not satisfy statement P This

23bf821d-0498-4ff1-a5ce-38f47523dda1-1
00:08:14.300 --> 00:08:15.960
gives a counterexample.

2d85c59d-6fa3-407e-b025-69343a2b34b2-0
00:08:16.240 --> 00:08:23.681
This gives an example of a function that
disproves statement 2 being sufficient

2d85c59d-6fa3-407e-b025-69343a2b34b2-1
00:08:23.681 --> 00:08:24.240
for P.

92c4cac1-bd67-400e-a74a-39ff6b00fe2d-0
00:08:24.400 --> 00:08:27.640
That means statement 2 is not sufficient
for P.

601384d8-bf1a-4b63-9551-13111286d009-0
00:08:27.640 --> 00:08:29.040
We've answered that?

f8720b75-fd3e-49ff-9afb-a4176097f1dd-0
00:08:29.320 --> 00:08:33.920
And so I'm going to put an X there so
that I can track as I go along.

4d315db2-02b3-4041-a169-3485f304b265-0
00:08:34.320 --> 00:08:37.600
Now I'm going to clear the board and we
can think about statement 3.

566919d8-d2e6-4710-bd4f-0d7b61fdb3e6-0
00:08:42.360 --> 00:08:47.200
Now let's think about statement 3 and
whether statement 3 is sufficient for Pi.

7116edeb-fc41-413a-b6d1-3b455570be04-0
00:08:48.480 --> 00:08:53.320
Want to start by focusing on this
condition here.

1cf5902e-0fb4-4d3e-ada7-d487265c9c67-0
00:08:54.800 --> 00:08:59.337
If we have two values here,
we're looking at the value of F of U and

1cf5902e-0fb4-4d3e-ada7-d487265c9c67-1
00:08:59.337 --> 00:09:04.729
the value of F of V and we're told that
their product is negative or less than 0,

1cf5902e-0fb4-4d3e-ada7-d487265c9c67-2
00:09:04.729 --> 00:09:10.187
then that just tells us that they have to
have opposite signs or they have to have

1cf5902e-0fb4-4d3e-ada7-d487265c9c67-3
00:09:10.187 --> 00:09:11.240
different signs.

535825ab-f120-4354-a959-0bed45c6d736-0
00:09:11.600 --> 00:09:16.941
So if F of U is positive,
this condition here forces F of V to have

535825ab-f120-4354-a959-0bed45c6d736-1
00:09:16.941 --> 00:09:18.120
to be negative.

d99b1afe-7760-4c4b-b840-c3ecbf041f67-0
00:09:18.480 --> 00:09:23.280
If F of V was positive,
then F of U would have to be negative.

ae9561c5-df67-4943-b20e-d6da271efd4c-0
00:09:23.560 --> 00:09:26.700
Let's think about an example to
illustrate this,

ae9561c5-df67-4943-b20e-d6da271efd4c-1
00:09:26.700 --> 00:09:29.520
and we'll think about our coordinate
plane.

f753ed9e-cc2d-4350-b5da-90a4a912795b-0
00:09:29.960 --> 00:09:36.400
Let's say that our value of U was there,
and let's say UF of U is negative.

95927433-6b1f-4711-ad59-51a3ab30fb79-0
00:09:36.720 --> 00:09:41.040
So let's put F of U around about here.

340eff1a-aced-49b1-828b-9a1e91167e88-0
00:09:42.160 --> 00:09:45.669
Well,
this condition forces F of V to be

340eff1a-aced-49b1-828b-9a1e91167e88-1
00:09:45.669 --> 00:09:46.440
positive.

a4eb849f-7365-4c88-8aaf-26db9159a6ff-0
00:09:46.680 --> 00:09:51.909
So if V is somewhere over here,
then F of V would have to be somewhere

a4eb849f-7365-4c88-8aaf-26db9159a6ff-1
00:09:51.909 --> 00:09:52.720
along here.

f0324a00-6154-48ee-9b23-39657605538b-0
00:09:53.160 --> 00:09:54.520
Let's put it here.

58b5d457-89d4-46e5-bb37-580d01ab62d0-0
00:09:54.520 --> 00:09:57.120
So let's let this be F of V.

002f4f07-237f-49fd-b641-0da1aa9313f0-0
00:09:57.440 --> 00:10:02.701
And if we have a polynomial function that
has to pass through both of these points,

002f4f07-237f-49fd-b641-0da1aa9313f0-1
00:10:02.701 --> 00:10:05.520
then it must cross the X axis at least
once.

45776391-fec5-44be-aec3-7408dd5158c2-0
00:10:05.880 --> 00:10:12.120
Maybe we've got a linear function and
that would pass through the X axis once.

af33f19d-b102-41eb-950f-41a13b73b668-0
00:10:12.440 --> 00:10:17.020
Maybe we've got a cubic function which
could pass through the X axis up to three

af33f19d-b102-41eb-950f-41a13b73b668-1
00:10:17.020 --> 00:10:17.360
times.

6e3a655f-bcf9-4c13-927f-4174d8affea2-0
00:10:18.000 --> 00:10:20.491
And you know,
we could have different polynomial

6e3a655f-bcf9-4c13-927f-4174d8affea2-1
00:10:20.491 --> 00:10:21.000
functions.

7d68ab8a-31c3-4e54-b9cb-879cf7de77f7-0
00:10:21.320 --> 00:10:25.419
But in order for them to pass through
both of these points,

7d68ab8a-31c3-4e54-b9cb-879cf7de77f7-1
00:10:25.419 --> 00:10:29.040
they have to pass to cross the X axis at
least once.

26143356-8a7a-4763-9403-973dc082930c-0
00:10:29.280 --> 00:10:33.758
Therefore,
guaranteeing that F of X = 0 has at least

26143356-8a7a-4763-9403-973dc082930c-1
00:10:33.758 --> 00:10:35.280
one real solution.

cdbef9b3-0b42-45e0-94a3-7420aa95b428-0
00:10:35.800 --> 00:10:39.960
This means that statement 3 is sufficient
for P.

794f082d-825c-48aa-b430-700e1013b7c9-0
00:10:40.800 --> 00:10:42.000
So I'm going to take that.

5119acaa-f80b-4e87-b23b-9249e679f1f1-0
00:10:42.440 --> 00:10:48.110
And at this point I can now look at the
options I'm given to choose from and I'm

5119acaa-f80b-4e87-b23b-9249e679f1f1-1
00:10:48.110 --> 00:10:52.800
looking for a yes, no, yes,
which on here is option C, yes no yes.

04813007-02c8-45e5-bf50-d4bf59d75a84-0
00:10:53.080 --> 00:10:55.720
So option C is the answer to this
question.

8ea6d97c-0ab1-487b-903e-ed4c5cdfb5ca-0
00:10:56.280 --> 00:10:58.571
OK,
let's take some time to reflect on this

8ea6d97c-0ab1-487b-903e-ed4c5cdfb5ca-1
00:10:58.571 --> 00:10:59.040
question.

5a8c92d2-3db8-4f03-8315-74e59f1900d0-0
00:10:59.320 --> 00:11:04.365
The main thing I wanted to point out is
how valuable it is to be aware of what

5a8c92d2-3db8-4f03-8315-74e59f1900d0-1
00:11:04.365 --> 00:11:06.920
this piece of information is telling us.

a890846b-3af3-4d07-92c8-5719a4a0b48f-0
00:11:07.240 --> 00:11:10.754
Now, in this question,
we were told that we had two values

a890846b-3af3-4d07-92c8-5719a4a0b48f-1
00:11:10.754 --> 00:11:12.840
multiplied together to be negative.

b53df64d-ab6d-4d88-85f3-51207d9c0f13-0
00:11:13.720 --> 00:11:16.960
So F of U * F of V less than 0.

efd0dfc8-c69f-4e25-b7e4-64f45b26671a-0
00:11:17.280 --> 00:11:22.306
And that told us that the value of the
function at U and the value of the

efd0dfc8-c69f-4e25-b7e4-64f45b26671a-1
00:11:22.306 --> 00:11:25.160
function at V had to have different signs.

1d455662-0940-490e-bbd5-5930fb332a2d-0
00:11:25.400 --> 00:11:31.092
You might find yourself in a situation
where you are told 2 values,

1d455662-0940-490e-bbd5-5930fb332a2d-1
00:11:31.092 --> 00:11:37.706
let's keep calling them F of U * F of V
and we could be told that the value of

1d455662-0940-490e-bbd5-5930fb332a2d-2
00:11:37.706 --> 00:11:40.720
the product of 2 values is positive.

7baa6504-f61d-4804-a445-6fee35c62fb1-0
00:11:41.120 --> 00:11:46.560
And what this will be telling us is that
both of these have to have the same sign.

22a2eef1-7b9c-481f-b8e3-4bc5177f85fb-0
00:11:46.800 --> 00:11:49.660
So this would be telling us if F of U is
positive,

22a2eef1-7b9c-481f-b8e3-4bc5177f85fb-1
00:11:49.660 --> 00:11:51.680
then F of V has to also be positive.

28008357-374f-4766-ad5a-2315e0af3a23-0
00:11:52.000 --> 00:11:56.560
And if F of U is negative,
F of V has to also be negative.

2e104b31-e835-4e8e-b2e7-e6bc21cb79f0-0
00:11:56.560 --> 00:12:00.600
So look out for information like this or
conditions like this.

34726c64-54be-408a-b979-9fab723834ff-0
00:12:01.360 --> 00:12:04.120
And now we know what they tell us.