WEBVTT

b705717f-ed38-4a15-a4a7-4e258f42295c-0
00:00:05.360 --> 00:00:06.320
Hi, I'm Richard.

440dc438-2816-4899-adc7-f4a18d863c7f-0
00:00:06.320 --> 00:00:10.160
We're going to be looking at Tamura 2021
paper one question 4.

93307f9c-fb72-4358-a811-19e77a713e71-0
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Here it is.

12527fb2-223a-462d-a5d8-5f99268d569c-0
00:00:10.920 --> 00:00:17.530
It says find the minimum value of the
function and we have two to the two X -,

12527fb2-223a-462d-a5d8-5f99268d569c-1
00:00:17.530 --> 00:00:19.120
2 to the X + 3 + 4.

d04b9783-d957-4970-b249-750a0683c733-0
00:00:19.840 --> 00:00:23.320
So we can see that this is a rather
complicated looking expression.

0b06584c-2f28-4d4e-9592-219e22ba908f-0
00:00:23.600 --> 00:00:25.932
We can certainly see,
imagine that the different values of X

0b06584c-2f28-4d4e-9592-219e22ba908f-1
00:00:25.932 --> 00:00:27.960
we're going to get different values of
the function.

7a1423b4-19e4-490c-b101-4a214d8f56ea-0
00:00:27.960 --> 00:00:31.707
So asking us to think about the minimum
value we can achieve feels like a

7a1423b4-19e4-490c-b101-4a214d8f56ea-1
00:00:31.707 --> 00:00:32.720
reasonable question.

f8e2bc2a-9a0c-4352-ad7e-babf60fde3fc-0
00:00:34.040 --> 00:00:38.914
The way we're going to attack this is
we're going to see that this expression

f8e2bc2a-9a0c-4352-ad7e-babf60fde3fc-1
00:00:38.914 --> 00:00:41.851
has actually got the structure of a
quadratic,

f8e2bc2a-9a0c-4352-ad7e-babf60fde3fc-2
00:00:41.851 --> 00:00:46.600
and we can use the fact that we could
find minimum values for quadratics to

f8e2bc2a-9a0c-4352-ad7e-babf60fde3fc-3
00:00:46.600 --> 00:00:49.600
find the minimum value for this
expression, OK.

5ca9e548-aa0b-44c6-9f19-9572d4120013-0
00:00:49.600 --> 00:00:52.781
So in order to see that,
we're going to look at each of these

5ca9e548-aa0b-44c6-9f19-9572d4120013-1
00:00:52.781 --> 00:00:55.040
terms in turn,
starting with two to the 2X.

c8adf9c0-600b-4f8f-adff-9df40f4ffb6e-0
00:00:55.600 --> 00:00:57.120
So we have two to the 2X.

fe3b769d-049d-4c9d-a5e0-6556f330b650-0
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We could write that in another way.

0dae4b73-1a44-4491-afa1-d827c03d3afb-0
00:00:59.120 --> 00:01:02.840
We could write that as 2 ^2,
all to the power of X.

3026c7a0-232f-4410-9943-687527e0f4d4-0
00:01:02.840 --> 00:01:06.160
So there's a law of indices which tells
us these two things are the same.

4fccabad-8e9e-47d8-92c7-06638cb2db25-0
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We can just multiply these two together
to get this power.

5ea23d31-6906-4a77-8744-90a13c643dbc-0
00:01:09.440 --> 00:01:14.280
Or we could use the same rule to write
that as two to the X ^2.

b66d9428-ad6c-4351-851f-62c9c0aea93f-0
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Now we can see that the first term is
actually the square of an unknown.

bc658108-f3fd-4b10-80d1-3388405f90d6-0
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It's the square of 2 to the X OK,
moving along to the second term,

bc658108-f3fd-4b10-80d1-3388405f90d6-1
00:01:22.473 --> 00:01:26.960
ignoring the - just for a second,
we have two to the X + 3 there.

7776cbf2-cbe1-4cb4-b6c3-a706a285614b-0
00:01:27.560 --> 00:01:31.742
We can write this as two to the X * 2 to
the three,

7776cbf2-cbe1-4cb4-b6c3-a706a285614b-1
00:01:31.742 --> 00:01:34.960
and we can write that as 8 * 2 to the X.

01accd8b-81c3-448b-a569-25b4c499d64a-0
00:01:35.680 --> 00:01:40.880
So now we see that the second term here
is actually a multiple of 2 to the X.

1c2483c2-2495-414a-bc65-ea94c559757e-0
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We have the first term is the square of 2
to the X.

71cc4397-570e-4087-bb4e-461fe47e7527-0
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The second term is a constant multiple of
2 to the X, and then we have a constant.

f046c07c-4452-47ed-b399-9861c11a85a1-0
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So we can see that this expression
actually has the structure of a quadratic.

117e0362-1499-46af-87e6-b7323baf2193-0
00:01:54.400 --> 00:01:58.532
So now we're going to move back over here
and we're going to exploit that to find

117e0362-1499-46af-87e6-b7323baf2193-1
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the minimum value.

b97905e9-fbe0-4ca6-a823-03c8a61c21f2-0
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So the first thing we're going to do is
we're going to replace our expression

b97905e9-fbe0-4ca6-a823-03c8a61c21f2-1
00:02:03.400 --> 00:02:04.560
using the ideas over here.

b8532904-6a20-4fd6-b564-564af65c9e21-0
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So 2 to the 2X is becoming 2 to the X ^2.

39a028cc-b2a1-4a98-9b44-f04c026f3f6d-0
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This term is becoming -8 * 2 to the X.

d886996c-22d7-46a0-9dc3-9c22cd9dbb12-0
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And then we carry over the +4.

50dfed9a-2367-4119-a8c1-76f34e95c267-0
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So again,
just to emphasise what we have got here

50dfed9a-2367-4119-a8c1-76f34e95c267-1
00:02:19.528 --> 00:02:21.400
has got the structure of a quadratic.

b0a02e10-e542-4455-aefc-c539b34bc351-0
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We have two to the X ^2.

251bbc56-1306-4c19-8286-857e8f188196-0
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We have a constant times 2 to the X and
we have a + 4.

a2167730-9059-4331-a55e-37232bea6603-0
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So it's like 2 to the X is in the role of
the normal variable in a quadratic

a2167730-9059-4331-a55e-37232bea6603-1
00:02:31.410 --> 00:02:32.040
expression.

9850951e-a034-4259-a76a-282f00cdec81-0
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So if we had a quadratic expression,
quadratic expression,

9850951e-a034-4259-a76a-282f00cdec81-1
00:02:35.963 --> 00:02:39.080
we might complete the square to find the
minimum value.

977f520c-9169-4026-aae9-bf2c1358f859-0
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So that's what we're going to do here to
complete the square.

8687dd93-6d38-448e-bed2-90b5048f14cf-0
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I'm going to write down the variable,
which is kind of two to the X for us.

003864d9-bdd0-4989-8289-f421eda639c0-0
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We're going to halve the coefficient on
this term, which will give us a -4,

003864d9-bdd0-4989-8289-f421eda639c0-1
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and we imagine squaring them.

2164d2e8-676a-41d6-bd78-d57e6eaa7746-0
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So if you were to square this,
we would pick up two to the X ^2,

2164d2e8-676a-41d6-bd78-d57e6eaa7746-1
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which is this thing.

00ba7338-4819-47d1-b8cc-067084bc4519-0
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We would pick up -4 * 2 to the X twice,
which will give us this term.

5b817901-62f2-4fcd-8510-9bf3a68b53ce-0
00:03:04.760 --> 00:03:08.320
And we would also pick up -4 * -, 4,
which is plus 16.

8850392a-61d1-4b14-afb7-95af4f3a298c-0
00:03:08.600 --> 00:03:13.586
Now we want equality with this line,
so we need to turn that plus 16 into a +

8850392a-61d1-4b14-afb7-95af4f3a298c-1
00:03:13.586 --> 00:03:16.080
4, and we'll do that by subtracting 12.

21e19ccf-f235-4696-9e4f-fcf5db79c2e5-0
00:03:16.520 --> 00:03:19.960
So now I can say this really is equality
here with the line above.

df581c43-bc66-459d-802b-7e41512d3b93-0
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OK, so if we look at this expression now,
imagine we were evaluating it at some

df581c43-bc66-459d-802b-7e41512d3b93-1
00:03:26.976 --> 00:03:27.760
value of X.

852e5236-ca50-47e0-bdec-31a9f243563d-0
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Well,
the value of this will always be greater

852e5236-ca50-47e0-bdec-31a9f243563d-1
00:03:30.956 --> 00:03:34.600
than or equal to 0 because it's the
square of some real number.

e41b1abb-9e88-4b17-9948-7c9bc9b3a371-0
00:03:35.320 --> 00:03:37.600
And this will always evaluate to -12.

a4951ced-184d-4f87-b5c6-00c7975ee33b-0
00:03:38.120 --> 00:03:42.252
So when we evaluate this at some X,
we'll get -12 plus something which is

a4951ced-184d-4f87-b5c6-00c7975ee33b-1
00:03:42.252 --> 00:03:43.760
greater than or equal to 0.

5d21a894-f69f-4def-be05-2b9378d06ceb-0
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So the smallest possible value we might
ever get is -12.

92d9991c-be53-4ffb-a00c-587f4a57a5ef-0
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We just need to be a little bit quite
careful and convince us that we could

92d9991c-be53-4ffb-a00c-587f4a57a5ef-1
00:03:53.173 --> 00:03:56.012
really get -12 So in order to get -12 for
this,

92d9991c-be53-4ffb-a00c-587f4a57a5ef-2
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we would need to get 0 for this part.

cb93e847-04dd-443b-962e-b346212c493a-0
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And because this is the square of
something,

cb93e847-04dd-443b-962e-b346212c493a-1
00:04:01.297 --> 00:04:04.360
we would need to be squaring 0 itself in
order to get 0.

6996ee86-b395-48c2-8f02-3f63c117e177-0
00:04:05.080 --> 00:04:07.400
So can we make this bracket into zero?

9da89a6f-aa64-4c43-b1c5-b24f8dcc5b59-0
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Well, yes, I think if we take X = 2 here,
we would have 4 -, 4,

9da89a6f-aa64-4c43-b1c5-b24f8dcc5b59-1
00:04:11.139 --> 00:04:12.600
and that would give us 0.

5af930e1-6c3f-4a33-8526-63ff0f66c378-0
00:04:12.840 --> 00:04:21.533
So we can say when X = 2,
the expression evaluates to -12,

5af930e1-6c3f-4a33-8526-63ff0f66c378-1
00:04:21.533 --> 00:04:30.080
and so that will be the minimum value it
can be achieved.

dbe111b9-a208-4877-9850-d1d0b32b7887-0
00:04:30.120 --> 00:04:33.480
And we look over our answers and we can
see -12 here as B.

81dfd955-1533-4132-8a92-e2f3519b1bcf-0
00:04:34.720 --> 00:04:37.335
Okay,
let's reflect on what we may take away

81dfd955-1533-4132-8a92-e2f3519b1bcf-1
00:04:37.335 --> 00:04:38.440
from this question.

5175c923-9495-4d18-89e0-b6341dea29af-0
00:04:39.960 --> 00:04:43.073
I guess when you approach this,
the first thing you see is that you're

5175c923-9495-4d18-89e0-b6341dea29af-1
00:04:43.073 --> 00:04:44.520
required to find a minimum value.

71d273eb-a902-4a97-a13e-581197934a15-0
00:04:44.600 --> 00:04:47.391
And that might get you thinking about
what are the,

71d273eb-a902-4a97-a13e-581197934a15-1
00:04:47.391 --> 00:04:50.880
what are the methods I have to find
minimum values of functions.

ef66acad-ae86-4ac2-9453-9bd0e1c0c083-0
00:04:51.120 --> 00:04:54.653
One of them will be differentiation,
but this looks kind of tricky to

ef66acad-ae86-4ac2-9453-9bd0e1c0c083-1
00:04:54.653 --> 00:04:55.360
differentiate.

b583fed5-9789-4463-a139-521b480173e5-0
00:04:56.800 --> 00:05:00.640
And we also know that we can use the
complete and the square idea for a

b583fed5-9789-4463-a139-521b480173e5-1
00:05:00.640 --> 00:05:03.520
quadratic to find the minimum value of
the quadratic.

18f82ba8-ac10-4277-a326-c614864ec15b-0
00:05:03.680 --> 00:05:06.080
So you might start by thinking about
those things.

641a4b4d-b5e0-4b2a-b0e7-e1e2fc1602f4-0
00:05:07.240 --> 00:05:12.574
The next thing I spotted in this is this
idea that this two to the 2X is actually

641a4b4d-b5e0-4b2a-b0e7-e1e2fc1602f4-1
00:05:12.574 --> 00:05:13.160
a square.

56a6218c-7271-4696-a716-ef98ffc6d1c6-0
00:05:13.480 --> 00:05:20.615
So you'll remember that we saw that two
to the 2X can be written as two to the X

56a6218c-7271-4696-a716-ef98ffc6d1c6-1
00:05:20.615 --> 00:05:20.880
^2.

6ec13234-c5fe-46df-92bd-ef2b806083ad-0
00:05:21.560 --> 00:05:25.554
So this is something you might want to
kind of bank or something to remember for

6ec13234-c5fe-46df-92bd-ef2b806083ad-1
00:05:25.554 --> 00:05:26.640
this kind of question.

b18c51e4-8595-49bf-999c-aa4ca2d8d583-0
00:05:27.080 --> 00:05:30.558
So if you ever you have two to the
something to the power of 2 times

b18c51e4-8595-49bf-999c-aa4ca2d8d583-1
00:05:30.558 --> 00:05:31.920
something, that's a square.

0a97dafa-4164-4eec-ac85-bdd7a13d3259-0
00:05:31.920 --> 00:05:36.557
So of course, 4 to the 2X4 to the X ^2,
or for example,

0a97dafa-4164-4eec-ac85-bdd7a13d3259-1
00:05:36.557 --> 00:05:40.532
2 to the power of 3X is actually 2 to the
X ^3,

0a97dafa-4164-4eec-ac85-bdd7a13d3259-2
00:05:40.532 --> 00:05:47.240
and you might be able to get a cubic kind
of expression from that which might be

0a97dafa-4164-4eec-ac85-bdd7a13d3259-3
00:05:47.240 --> 00:05:48.400
useful to you.

b6bd5a22-2e5a-4d1d-b580-212ea501da1e-0
00:05:49.000 --> 00:05:51.833
The four here is a little bit of a trap
potentially,

b6bd5a22-2e5a-4d1d-b580-212ea501da1e-1
00:05:51.833 --> 00:05:56.272
because it might lead you to think about
trying to take a factor of 2 out of this,

b6bd5a22-2e5a-4d1d-b580-212ea501da1e-2
00:05:56.272 --> 00:05:59.319
noticing that you've got a 2 in each of
the terms there.

3f9e746d-4b3f-46f9-b328-9a0b60db123d-0
00:06:00.280 --> 00:06:01.760
I don't think that would head anywhere.

f91be82c-ea76-489c-8a6a-95f0dec22fa7-0
00:06:01.760 --> 00:06:05.792
So that I think that's been deliberately
set as perhaps a little bit of a trap to

f91be82c-ea76-489c-8a6a-95f0dec22fa7-1
00:06:05.792 --> 00:06:07.760
get you to steer in the wrong direction.

6a4027d1-0ec2-4338-ad36-362a30262ae7-0
00:06:07.760 --> 00:06:12.391
Maybe the next thing I want to reflect on
is the quadratic nature of this

6a4027d1-0ec2-4338-ad36-362a30262ae7-1
00:06:12.391 --> 00:06:13.080
expression.

f1c155dd-dff1-4d08-baf7-dd0c0367358d-0
00:06:13.320 --> 00:06:18.027
So we dealt with this by sticking with
two to the X here and working with two to

f1c155dd-dff1-4d08-baf7-dd0c0367358d-1
00:06:18.027 --> 00:06:19.480
the X in this expression.

14d68b52-18b1-48c6-a489-64e727e01ab7-0
00:06:19.840 --> 00:06:22.354
When you get this guy's quadratics like
this,

14d68b52-18b1-48c6-a489-64e727e01ab7-1
00:06:22.354 --> 00:06:25.360
you may be more used to giving this a new
letter name.

1dfb6f67-8514-4548-b1c2-483b958b8fe8-0
00:06:25.360 --> 00:06:31.280
So you might say a = 2 to the X and then
rewrite this as a ^2 -, 8 A plus four.

3a1891b8-c2ea-41f4-9f8b-40d3eb1ab57e-0
00:06:31.600 --> 00:06:33.720
And then all this would then be in terms
of A.

9b46cbd6-e9a9-48f1-ba6c-bd9bf2bcd427-0
00:06:34.480 --> 00:06:34.880
You do.

6cbf2dd0-fffd-4ca3-9b7e-f749d37d9203-0
00:06:34.880 --> 00:06:37.900
If you do that,
then you do need to remember that you

6cbf2dd0-fffd-4ca3-9b7e-f749d37d9203-1
00:06:37.900 --> 00:06:41.200
need to check that that a could actually
take the value 4.

b00bb8a4-cdd5-41df-a215-e1a011e37e9b-0
00:06:41.200 --> 00:06:44.240
So you need to remember that a is
actually 2 to the X in that little bit of

b00bb8a4-cdd5-41df-a215-e1a011e37e9b-1
00:06:44.240 --> 00:06:45.280
interpretation at the end.

42b57c84-06ee-40eb-bfb7-74c84005940e-0
00:06:46.400 --> 00:06:49.540
I would encourage you,
if you feel confident enough and you can

42b57c84-06ee-40eb-bfb7-74c84005940e-1
00:06:49.540 --> 00:06:52.141
work towards this,
to actually stick in terms of the

42b57c84-06ee-40eb-bfb7-74c84005940e-2
00:06:52.141 --> 00:06:54.840
original variables and don't introduce a
new variable.

4cbbf5c6-a99b-4cd8-afca-8ff6f0718717-0
00:06:55.120 --> 00:06:59.946
That's a kind of a nice little step up in
terms of you feeling confident with these

4cbbf5c6-a99b-4cd8-afca-8ff6f0718717-1
00:06:59.946 --> 00:07:01.440
methods, and often it can.

36f57d69-5c6c-4b59-9ef7-12794d4c782b-0
00:07:01.440 --> 00:07:06.088
Just make sure you don't fall into traps
by forgetting what your substitutions are

36f57d69-5c6c-4b59-9ef7-12794d4c782b-1
00:07:06.088 --> 00:07:07.040
and their nature.