WEBVTT

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Hi, I'm Fiona.

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Let's have a look at Timura 2021 paper
one and question 12.

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We are told that the minimum value of the
function X to the power of 4 -,

0e11e87f-2093-4ed3-9ab3-e978b1cd6efc-1
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P ^2 X ^2 is -9 and that P is a real
number.

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And we're asked to find the minimum value
of the function X ^2 minus PX +6.

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And then we're given options 8F to choose
from once we get to that point.

93ecba73-9c67-47be-bbd8-9ec234230a98-0
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Now when we think about a quadratic like
this,

93ecba73-9c67-47be-bbd8-9ec234230a98-1
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X ^2 minus PX +6 and where or how we find
its minimum value,

93ecba73-9c67-47be-bbd8-9ec234230a98-2
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then we start to think about completing
the square.

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So I'm going to start by completing the
square on this quadratic.

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Here I have X ^2 minus PX +6.

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Now to complete the square on this,
I'm going to have X -,

0336ab2e-1cde-40b2-80ee-a37bb4f64f18-1
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P / 2 all squared in a bracket.

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And then when I multiply this out as a
constant term, I get a positive P ^2 / 4.

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And so I need a negative P ^2 / 4 in
order to balance out and keep these two

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lines equal.

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And then I'm going to bring down this +6
as well.

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So in order for this particular function
to have A to be at its minimum value,

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then I can see using this completed
square form that this part of the

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function is always going to be positive
because it's squared.

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And so the function itself will take its
minimum value when what's in the bracket

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equals 0.

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And so because of that,
then I look here to find the minimum

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value of my function.

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So this function takes a minimum value of
-P ^2 / 4 + 6,

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but I don't know what P is yet.

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And so now I'm going to go back to the
first line of the question and use this

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information over here in order to get a
sense of the value of P.

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Now when I look at this function X ^2,
not X ^2 X to the power of 4 -, P ^2,

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X ^2.

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This is a polynomial with degree 4,
but I can also think of it as a quadratic

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in X ^2.

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So I can think of this as the same as X
^2 ^2 -, P ^2, X ^2.

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And so let me complete the square on this
function considering as a quadratic in X

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^2 and then that will help me to find out
the value of P.

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So in my bracket this time I have X ^2
and I have minus P ^2 / 2.

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And then this is all squared.

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And then I when I multiply this out,
I would get a constant term of plus P to

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the power of 4 / 4.

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And so I need a -,
P to the power of 4 / 4 in order to keep

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these two lines equal.

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And here I don't,
I didn't start with a constant term.

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So it's just minus P to the power of 4 /
4.

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And in the same way that for this
function it took,

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we could find its minimum value by
considering when this particular

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component was zero,
I can do the same over here.

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So when what's in the bracket here is 0,
this function will take a minimum value.

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So that tells me that minus P to the
power of 4 / 4 is the value at which this

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function has a minimum,
and that is equal to -9.

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So minus P to the power of 4 / 4 = - 9.

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And that tells me that P to the power of
4 = 9 * 4, which is 36.

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And now I just look over here because I
don't necessarily need P but I need P ^2

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and so I can take the square root of both
sides here and I get that P ^2 = ± sqrt

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36.

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That means that P ^2 = ± 6.

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Now if I take -6 as the value for P ^2,
then the then that will not be the lowest

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value that this can take here.

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And so I'm assuming that I should take a
value of +6 for P ^2.

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But I need to just give a little bit of
extra thinking to this because I'm told

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here that P is a real number.

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And so if I did think if I did take P ^2
to be -6,

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then that would mean P would not be a
real number.

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Because in order to get the actual value
of P,

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I'd need to find the square root of a
negative number.

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And so that tells me and confirms to me
that I should take the positive value for

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P ^2, that P ^2 = a positive 6.

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So substituting that in here,
I get -6 / 4 + 6 and that's going to be

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-3 / 2 + 6.

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And six is 12 halves.

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Take away three of those gives me 9
halves.

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And so 9 / 2 is the value and the minimum
value of the function that I'm looking

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for.

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And having a look over at my options,
I can see that option E gives a value of

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9 / 2.

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And that's the answer to this question.