WEBVTT

4c65d5e1-442e-4c2a-a27c-4f51a765a5ab-0
00:00:05.320 --> 00:00:06.280
Hi I'm Fiona.

b317255b-1e4d-4008-b1a6-2c8b2e078f5f-0
00:00:06.680 --> 00:00:12.440
Let's have a look at Tamua 2021 paper one
and question 11.

90cb22ce-cd4a-496b-97ca-40ee3dab845f-0
00:00:12.960 --> 00:00:19.399
The function F is given by F of X = X to
the power of 1/7 on X ^2 -, X + 1,

90cb22ce-cd4a-496b-97ca-40ee3dab845f-1
00:00:19.399 --> 00:00:26.347
and we are asked to find the fraction of
the interval when X takes values between

90cb22ce-cd4a-496b-97ca-40ee3dab845f-2
00:00:26.347 --> 00:00:30.160
zero and one for which F of X is
decreasing.

c3af9456-0c9c-4a35-8f19-6582d67e06d8-0
00:00:30.400 --> 00:00:34.560
We're giving some options to choose from
once we get to that point in the question.

0991d278-4f70-4532-b6c7-2fcf44d33d4e-0
00:00:35.280 --> 00:00:40.656
Now when we're considering polynomial
functions and for which parts of their

0991d278-4f70-4532-b6c7-2fcf44d33d4e-1
00:00:40.656 --> 00:00:45.964
domain they are increasing or decreasing,
one of the things we can do is to

0991d278-4f70-4532-b6c7-2fcf44d33d4e-2
00:00:45.964 --> 00:00:47.640
consider the derivative.

6f0c0cf2-b319-486a-b5ab-4715b19e8c83-0
00:00:48.040 --> 00:00:52.957
Because the for the parts of a function
on which it is decreasing,

6f0c0cf2-b319-486a-b5ab-4715b19e8c83-1
00:00:52.957 --> 00:00:56.480
the gradient of that function will be
negative.

d236555f-e893-4c1e-b58c-08613fcdd733-0
00:00:56.960 --> 00:01:00.246
And for the parts of a function on which
it is increasing,

d236555f-e893-4c1e-b58c-08613fcdd733-1
00:01:00.246 --> 00:01:02.920
the gradient of that function will be
positive.

0a86e464-4db5-42e0-8359-84e6bac39ad1-0
00:01:03.160 --> 00:01:05.880
So I'm going to start by differentiating
this function.

e125ec80-d936-473a-9e79-9ad3f357005c-0
00:01:06.760 --> 00:01:10.824
Now having a look at the function,
we could use the product rule to

e125ec80-d936-473a-9e79-9ad3f357005c-1
00:01:10.824 --> 00:01:11.840
differentiate it.

44c6661a-66ef-497d-844c-dfd8a1220eab-0
00:01:12.080 --> 00:01:17.534
What I'm going to do is just multiply out
this bracket so that I can get F of X as

44c6661a-66ef-497d-844c-dfd8a1220eab-1
00:01:17.534 --> 00:01:22.200
a function comprised of three terms and
then differentiate from there.

3c331e82-cef5-4477-9ec8-550f52ce28aa-0
00:01:23.160 --> 00:01:28.671
So adding my powers,
my first term will be X to the power of

3c331e82-cef5-4477-9ec8-550f52ce28aa-1
00:01:28.671 --> 00:01:35.266
1/7 + 2, which is 14 sevenths,
so that will give me X to the power of 15

3c331e82-cef5-4477-9ec8-550f52ce28aa-2
00:01:35.266 --> 00:01:36.080
sevenths.

9225a16b-097d-4175-bce7-e6784b4f6097-0
00:01:36.560 --> 00:01:42.373
Then my next term will be minus X to the
power of 1/7 + 1, which is 7 sevenths,

9225a16b-097d-4175-bce7-e6784b4f6097-1
00:01:42.373 --> 00:01:46.080
so that will give me X to the power of 8
sevenths.

ece4c65b-5cb4-496e-90c2-aa6483ccd209-0
00:01:46.440 --> 00:01:52.560
And then my final term will be a positive
X to the power of 1 / 7.

9468957b-7456-4c22-9e18-2a455584a5d0-0
00:01:53.320 --> 00:01:55.680
Now I'm going to go ahead and
differentiate this function.

722f823c-81cf-4104-b724-6dceeeca05b5-0
00:01:56.520 --> 00:02:07.534
So F-X will be 15 / 7 * X and then
reducing this power by one will give me a

722f823c-81cf-4104-b724-6dceeeca05b5-1
00:02:07.534 --> 00:02:09.680
power of 8 / 7.

27924d7e-6ecf-4f59-93cd-390921a53d6e-0
00:02:10.160 --> 00:02:15.320
Then I will have -8 / 7 * X to the power
of.

89f8e27f-b765-4e1a-b32a-b0a5ab7d7617-0
00:02:15.680 --> 00:02:19.320
Reducing this by one will give me a power
of 1 / 7.

7767816c-9864-4ed9-9659-8e8a3e3c09d2-0
00:02:19.640 --> 00:02:27.890
And then my final term in the derivative
will be plus X to the power of -6

7767816c-9864-4ed9-9659-8e8a3e3c09d2-1
00:02:27.890 --> 00:02:31.960
sevenths, reducing that power by one.

a933d11e-e323-4842-84c3-5b17fd9f6e6e-0
00:02:32.080 --> 00:02:40.240
Oh, I also need to multiply by 1/7,
so 1/7 * X to the power of -6 / 7.

e354033e-f842-4f68-932c-401d4411109f-0
00:02:41.040 --> 00:02:42.920
OK, now I've got my derivative function.

17ca8ceb-9e27-4a90-9ee4-dd6cfb7e6bae-0
00:02:43.160 --> 00:02:48.478
In order to investigate the intervals on
which this is positive or negative,

17ca8ceb-9e27-4a90-9ee4-dd6cfb7e6bae-1
00:02:48.478 --> 00:02:53.520
which will tell us the intervals on which
F is increasing or decreasing.

943413bb-eaae-4f37-a7da-b6de214e9d9b-0
00:02:54.520 --> 00:03:00.608
I want to factorise this function or this
derivative function and then I can get

943413bb-eaae-4f37-a7da-b6de214e9d9b-1
00:03:00.608 --> 00:03:05.720
more of a handle on the areas for which
it is positive or negative.

765b50cd-d665-4cef-8fc6-9fd034895274-0
00:03:06.000 --> 00:03:09.296
So looking at this,
the first thing I want to do is take out

765b50cd-d665-4cef-8fc6-9fd034895274-1
00:03:09.296 --> 00:03:13.456
a factor of 1/7 and I'm also going to
take out a factor of X to the power of

765b50cd-d665-4cef-8fc6-9fd034895274-2
00:03:13.456 --> 00:03:15.240
1/7 and see where that leaves me.

ac6971a4-1504-44f8-a337-e99a091fd8c6-0
00:03:16.760 --> 00:03:28.250
So I will have 1/7 * X to the power of
1/7 and in my brackets I will have 15 X8

ac6971a4-1504-44f8-a337-e99a091fd8c6-1
00:03:28.250 --> 00:03:29.400
seventh.

6d358728-4156-4485-a355-79d8cc2c3c01-0
00:03:29.400 --> 00:03:39.653
Take away 1/7 is 7 sevenths which is 1 -
8 * 1 because I've taken out a factor of

6d358728-4156-4485-a355-79d8cc2c3c01-1
00:03:39.653 --> 00:03:45.280
X to the power of 1 / 7 + X to the power
of.

40399eaf-bc83-4043-bdec-eb443a52c5a1-0
00:03:45.320 --> 00:03:50.939
Well taking out a factor of 1 / 7 this
would leave me with a power of -7

40399eaf-bc83-4043-bdec-eb443a52c5a1-1
00:03:50.939 --> 00:03:54.480
sevenths which is -1 so X to the power of
-1.

3552c73d-abbe-4107-b2a5-01519a213a75-0
00:03:55.280 --> 00:04:01.893
Now I'm looking at this and if I do a
further factorization then I can get this

3552c73d-abbe-4107-b2a5-01519a213a75-1
00:04:01.893 --> 00:04:08.424
factor in the form of a quadratic and
then see if I can factorise further from

3552c73d-abbe-4107-b2a5-01519a213a75-2
00:04:08.424 --> 00:04:08.920
there.

b6306485-7fc3-4ad1-8787-ad5dd6313a0f-0
00:04:09.200 --> 00:04:14.514
Now in order to get this component in the
form of a of a quadratic,

b6306485-7fc3-4ad1-8787-ad5dd6313a0f-1
00:04:14.514 --> 00:04:21.000
what I want to do is multiply across by a
power of X and that's the same as taking

b6306485-7fc3-4ad1-8787-ad5dd6313a0f-2
00:04:21.000 --> 00:04:22.720
a factor of 1 / X out.

64e5b5ea-5fd9-4660-9fc0-b54096cbf34c-0
00:04:22.960 --> 00:04:24.120
So that's what I'm going to do.

0d39b7a6-1357-4434-ae6d-8c91bb27d776-0
00:04:24.840 --> 00:04:31.674
If I take a factor of 1 / X out,
then outside the brackets I'm left with 1

0d39b7a6-1357-4434-ae6d-8c91bb27d776-1
00:04:31.674 --> 00:04:35.320
/ 7 X to the power of 1/7 -, 7 sevenths.

5acd4679-0895-44a0-b3ce-4c7afb284c10-0
00:04:35.560 --> 00:04:39.680
So that's going to give me minus a power
of -6 sevenths.

431bf0e9-1ad7-4473-9577-dc47186cf8f9-0
00:04:40.120 --> 00:04:47.160
And then this will become fifteen X ^2 -,
8 X plus one.

8f62a219-a7ee-4b80-ae10-667e17c7abd5-0
00:04:47.880 --> 00:04:51.200
And now I've got a quadratic that I'm
hoping will factorise easily.

fb072a37-c36c-4e28-8267-59f410adbb04-0
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Let's have a look.

d688b88e-5a66-492a-bcb0-0504e14b9733-0
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So the next line would be 1/7 * X to the
power of -6 / 7.

a59cfdf6-74b4-4699-8e64-e09b3c3fda08-0
00:04:58.960 --> 00:05:06.137
Now this does factorise easily because 15
and factorises into 3 * 5 and three and

a59cfdf6-74b4-4699-8e64-e09b3c3fda08-1
00:05:06.137 --> 00:05:07.800
five adds to get 8.

506ea1f6-8ce8-47e0-9b96-445f6ca51c6f-0
00:05:08.080 --> 00:05:15.061
So if I have 5 X -1 and a three X -, 1,
then that's a factorization of this

506ea1f6-8ce8-47e0-9b96-445f6ca51c6f-1
00:05:15.061 --> 00:05:16.440
quadratic here.

629a16a3-180f-40d5-b3b1-7177c7b8c8d4-0
00:05:17.240 --> 00:05:22.462
So now that I have my derivative function
as a product of three factors,

629a16a3-180f-40d5-b3b1-7177c7b8c8d4-1
00:05:22.462 --> 00:05:26.897
that's what I want,
because I actually know how each of these

629a16a3-180f-40d5-b3b1-7177c7b8c8d4-2
00:05:26.897 --> 00:05:28.400
three factors behave.

ff60551e-f622-4091-91a7-3a5b656a4e61-0
00:05:28.960 --> 00:05:38.840
This factor is going to be equal to 0
when X is 1/5.

7235dca2-c17e-4e4c-859c-c11f5191eb4a-0
00:05:39.920 --> 00:05:45.720
This factor is going to be equal to 0
when X is 1/3.

42373bee-3bce-46e4-8b8a-bec60d0a2130-0
00:05:47.600 --> 00:05:52.412
And thinking about this,
this is the same as 1 / 7 to the power 7

42373bee-3bce-46e4-8b8a-bec60d0a2130-1
00:05:52.412 --> 00:05:54.600
* X to the power of 6 seventh.

f4f54bb8-4f00-4ab9-bd38-7f02c60def34-0
00:05:54.600 --> 00:05:59.532
So it's a reciprocal function,
and we're only considering values of X

f4f54bb8-4f00-4ab9-bd38-7f02c60def34-1
00:05:59.532 --> 00:06:03.760
between zero and one,
and so this part of the function will

f4f54bb8-4f00-4ab9-bd38-7f02c60def34-2
00:06:03.760 --> 00:06:06.720
just remain positive across that interval.

6a2d12e6-78aa-43b9-9886-3b1efdbeb58c-0
00:06:07.560 --> 00:06:13.542
Now in order to more clearly see the the
positivity or negativity of each of the

6a2d12e6-78aa-43b9-9886-3b1efdbeb58c-1
00:06:13.542 --> 00:06:17.974
factors of this function and therefore
the function itself,

6a2d12e6-78aa-43b9-9886-3b1efdbeb58c-2
00:06:17.974 --> 00:06:23.440
the derivative function itself,
I'm going to construct a sine table SIGN.

a14d056a-0f4a-4917-bc65-4e175967fda2-0
00:06:23.680 --> 00:06:27.862
And if you haven't seen this before,
then it can be a powerful tool to use

a14d056a-0f4a-4917-bc65-4e175967fda2-1
00:06:27.862 --> 00:06:29.480
when investigating functions.

2907a268-1461-4196-ac35-71ef185a4e67-0
00:06:45.280 --> 00:06:50.013
Now for my sine table,
I've got values of X running across the

2907a268-1461-4196-ac35-71ef185a4e67-1
00:06:50.013 --> 00:06:54.522
top of the table,
and the values of X that I've chosen have

2907a268-1461-4196-ac35-71ef185a4e67-2
00:06:54.522 --> 00:07:00.308
been determined based around these values
that I chose that I found earlier,

2907a268-1461-4196-ac35-71ef185a4e67-3
00:07:00.308 --> 00:07:03.839
and you'll see why when we get into the
table.

acdba0b3-e10d-4921-9c76-2ee8b087b86f-0
00:07:04.400 --> 00:07:09.992
And then the different components of the
function whose product makes up the

acdba0b3-e10d-4921-9c76-2ee8b087b86f-1
00:07:09.992 --> 00:07:12.680
function itself are listed down here.

b32eb026-1b33-4675-87fb-be4a3cc4742b-0
00:07:13.440 --> 00:07:17.809
What I'm going to do is for each of these
intervals or values of X,

b32eb026-1b33-4675-87fb-be4a3cc4742b-1
00:07:17.809 --> 00:07:21.600
I'm going to consider each of these parts
of the function.

5f980b4d-6675-4b4f-8f84-3bc237a0808f-0
00:07:21.840 --> 00:07:27.120
And then that will help me to investigate
the sign of the function overall.

72ada75b-c1bf-4f94-84cb-e988cdd7246c-0
00:07:27.120 --> 00:07:32.600
So I'll write here F-X because I want to
know the sign of my derivative function.

e2c8f4dd-623d-4724-b6cd-c152fb957c91-0
00:07:33.040 --> 00:07:39.245
So we already said that because this that
should be positive there because this is

e2c8f4dd-623d-4724-b6cd-c152fb957c91-1
00:07:39.245 --> 00:07:45.151
a reciprocal function and we are only
considering values of X between zero and

e2c8f4dd-623d-4724-b6cd-c152fb957c91-2
00:07:45.151 --> 00:07:47.319
one all across this interval.

a8ccf721-07a3-4f14-afb3-288d150fa2d4-0
00:07:47.600 --> 00:07:50.200
This part of the function is positive.

4d717327-e053-4812-8e9b-c8e9688b0c2d-0
00:07:50.600 --> 00:07:53.520
So I'm going to put a plus in to indicate
that.

951a5ec9-8224-4548-af3d-9fd72d1c9d52-0
00:07:55.320 --> 00:07:58.462
Now,
this part of the function we already said

951a5ec9-8224-4548-af3d-9fd72d1c9d52-1
00:07:58.462 --> 00:08:00.000
equals zero at X = 1/5.

7e2c0ca0-6932-4350-88a9-474943def2ad-0
00:08:01.360 --> 00:08:06.099
It's linear and it has a positive
coefficient of X and so for values of X

7e2c0ca0-6932-4350-88a9-474943def2ad-1
00:08:06.099 --> 00:08:11.415
less than 1/5 it's going to be negative,
and for values of X greater than 1/5 it's

7e2c0ca0-6932-4350-88a9-474943def2ad-2
00:08:11.415 --> 00:08:12.759
going to be positive.

4abf46f6-5f92-4798-b0a4-91aa79e1624b-0
00:08:16.480 --> 00:08:18.560
Now let's look at this component of the
function.

2db8fb17-00c0-4d69-967f-f5c4b980544e-0
00:08:18.720 --> 00:08:21.760
We said that it's zero when X = 1/3.

ab512ec6-c3eb-4bce-a3fc-a4a95aef9f9c-0
00:08:23.160 --> 00:08:28.120
It too is linear with a positive
coefficient of X and so for values of X

ab512ec6-c3eb-4bce-a3fc-a4a95aef9f9c-1
00:08:28.120 --> 00:08:33.692
less than 1/3 it's going to be negative
and for values of X greater than 1/3 it's

ab512ec6-c3eb-4bce-a3fc-a4a95aef9f9c-2
00:08:33.692 --> 00:08:35.120
going to be positive.

b8de304b-efaa-44f0-8d69-5ac62413afa5-0
00:08:36.720 --> 00:08:42.926
And so now I have my sine table mainly
filled out in such a way where I can

b8de304b-efaa-44f0-8d69-5ac62413afa5-1
00:08:42.926 --> 00:08:49.460
quickly have a look at whether F-X is
positive or negative on each of these and

b8de304b-efaa-44f0-8d69-5ac62413afa5-2
00:08:49.460 --> 00:08:50.440
values of X.

3fc082dc-9bc2-4931-9801-579cbed6d1ac-0
00:08:50.640 --> 00:08:54.897
So for this first interval,
I have a positive component multiplied by

3fc082dc-9bc2-4931-9801-579cbed6d1ac-1
00:08:54.897 --> 00:08:59.519
a negative component multiplied by a
negative component that will give me a

3fc082dc-9bc2-4931-9801-579cbed6d1ac-2
00:08:59.519 --> 00:09:02.560
positive value for the function on that
interval.

fe34f18c-cecd-4536-b523-4fba77fccbf7-0
00:09:03.240 --> 00:09:08.488
And the function will be 0 here because a
positive number times 0 times a negative

fe34f18c-cecd-4536-b523-4fba77fccbf7-1
00:09:08.488 --> 00:09:09.880
number will be 0 here.

7c28d09e-0945-4d4e-b100-188d8347d2ce-0
00:09:09.880 --> 00:09:13.880
I've got a positive number multiplied by
a positive number multiplied by a

7c28d09e-0945-4d4e-b100-188d8347d2ce-1
00:09:13.880 --> 00:09:14.360
negative.

c0b7724a-2606-462a-9e24-2d6255a75e60-0
00:09:14.600 --> 00:09:21.000
So that's going to be negative and here
my function will be 0 and here positive.

434bb201-ec07-4bdc-b27c-062e28b9601f-0
00:09:21.240 --> 00:09:27.972
So I can see that for this interval when
X is between 1/5 and 1/3,

434bb201-ec07-4bdc-b27c-062e28b9601f-1
00:09:27.972 --> 00:09:32.895
that is when my derivative function is
negative,

434bb201-ec07-4bdc-b27c-062e28b9601f-2
00:09:32.895 --> 00:09:37.920
which is when my original function is
decreasing.

5b47d830-12b2-4088-b667-668d4479f243-0
00:09:38.840 --> 00:09:44.387
And so now I know that this is the
interval within for values of X between

5b47d830-12b2-4088-b667-668d4479f243-1
00:09:44.387 --> 00:09:48.160
zero and one on which this function is
decreasing.

1f86426c-b5d4-45c4-aa7e-210d177b1871-0
00:09:48.480 --> 00:09:53.716
And just to wrap this question up,
I now need to find out what fraction of

1f86426c-b5d4-45c4-aa7e-210d177b1871-1
00:09:53.716 --> 00:09:56.160
this interval this part represents.

ca33a664-c529-47c5-8488-62477c60a2b9-0
00:09:57.000 --> 00:10:00.581
So to do that, I'm going to get 1/3,
take away 1/5,

ca33a664-c529-47c5-8488-62477c60a2b9-1
00:10:00.581 --> 00:10:04.920
and then put it all over the size of the
interval, which is 1.

703be7ef-9eb9-40d3-bd99-7496d72b3bcc-0
00:10:05.800 --> 00:10:12.320
And so 1/3 -,
1/5 is going to be 5 fifteenths -3

703be7ef-9eb9-40d3-bd99-7496d72b3bcc-1
00:10:12.320 --> 00:10:18.840
fifteenths,
which is going to be two fifteenths.

bce5a0c1-e1d8-4951-98da-a88a88bfa723-0
00:10:19.240 --> 00:10:25.168
So that's the fraction of the interval
for values of X between zero and one on

bce5a0c1-e1d8-4951-98da-a88a88bfa723-1
00:10:25.168 --> 00:10:27.720
which this function is decreasing.

fbf65bff-e0e0-4fef-958a-444f17bc071e-0
00:10:27.960 --> 00:10:33.478
And now I look at my options that are
available to me to choose an answer from

fbf65bff-e0e0-4fef-958a-444f17bc071e-1
00:10:33.478 --> 00:10:37.880
and I can see that option A gives the
value of two fifteenths.

46cfd850-e664-4dd4-83ba-a7a3b5ce0c13-0
00:10:38.120 --> 00:10:39.840
So that is the answer to this question.