Stanford Math Tournament Algebra Tiebreaker
okay for today's question it could be pretty hard but I have an easy solution for you guys so check
this out first we are given this cubic function and our goal is to pretty much find the interval
a b so that the sequence x, f(x), f(f(x)), and so on so on is going to be bounded for all X on this
interval and this question is from the Stanford math tournament the algebra breaker this year
first you might be wondering how come they are asking us for this sequence to be bounded why
not converge my guess for that is because if it's asking for us to get a convergence sequence that
sounds more like calculus instead of algebra so that's just my guess and you might also be
wondering how can I didn't write down the whole question whereby work on the board well the answer
for that is because my board is also bounded but anyway as always please pause the video and try
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solution and then later on I will show you guys the screenshot of the official solution so how
can we make this work well notice this sequence and this is actually called an iteration X and
you put it into the function and you put a result into the function again and so on so on so on and
there's always one kind of number that will make this pound it and that is the fixed point so what
exactly is that let me give you guys an example real quick so suppose we have a function f of x
and let's say this is equal to x square minus six take a look if I have x equals three then
this right here will work out very nicely but it's kind of boring because you'll see F of 3 is just
3 Square minus six of course that will give us three active worked out then we will do F of f
of 3. now we know F of 3 is 3 already so this is f of three and F of 2 we know that's equal to three
song this is always going to give us three so of course it's going to be bounded it's converge
right so that's that all right that's good so how can we use this idea to help us solve that
though well wouldn't it be nice if you can find the old infix points for this function
yes and in fact that's exactly all we have to do for this particular question all right
so now all we need to do is to find all the fixed points of this function first so we are looking at
x to the third power minus six x squared plus 25 over 2x minus 7 and we set this to be X and that's
just work this out real quick so minus example sets we get x to the third power minus six x
squared plus this minus 2 over 2 right so we get 23 over 2 x minus 7 that's equal to 0 and multiply
Everybody by two we get 2x plus the power minus 12 x squared plus 23x minus 14 and that's equal
to zero this is the cubic equation the best way to solve it of course is to use the cubic formula no
just kidding guess and check first look at the factors of 2 and then 14 and do the plus minus
but I'm going to delete that to you I'll just tell you note x equals 2 is a root feel free to use the
cubic formula though but don't do it all right that implies we will have x minus 2 as a factor
of this and then we can multiply that by another quadratic factor and to figure this out let's just
go ahead and use the synthetic division so write down the coefficients 2 negative 12 23 negative
14 and we put down a 2 right here first we bring down these two and then we do 2 times 2 and we
put that here which is 4. for synthetic division we add so this and that together we get negative
eight and then just repeat the process this times this which is negative 16 and then we add we get
7 and then we take this times that we get 14 this plus this is zero and she always gets zero because
we know that x minus 2 is a factor these right here are the coefficients and we will get 2 x
the power goes down by one so two x squared then minus 8. X plus seven okay from here we know that
2 is a root and for the other answers we just have to solve this quadratic equation and let's
just use the quadratic formula so we will have x equals negative B which is negative eight and
then plus well minus square root of B squared so negative a square minus 4AC a is 2 and C is seven
all divided by 2 times a so 2 times 2. and then let's just go and simplify this and I'll leave
that to you X is equal to 2 plus or minus square root of 2 over 2. now notice this is really cool
these are these three fixed point for this function and two it's in the middle of these
two numbers and because the question is asking us to find the interval there is an answer for
this and let me tell you the answer is in fact well the small number we have 2 minus square
root of 2 over 2 and then the other one two plus square root 2 over 2. this is the answer
of course you have to be careful with this
this is not always going to work for this kind of iterations but it does work for this and not let
me show you guys why this is super cool because we really don't have to do math for it we can just
draw some diagram but I always did here get some numbers so we can kind of understand this better
let's say I want to pick 1.5 for my x value which is about right here then I will have to get f of x
which is f of 1.5 and to do so I can actually just go to the curve and this right here will give me
1.625 and that's the Y value right and now I will have to do F of f of x in another word f of 1.625
how can we do it where the choices we really don't have to plug in 1.625 into f
check this out here is the point that we have now if we go to the left then that's the Y value 1.625
and if we go to the line this point here because Y is equal to X that will tell us
the corresponding x value down here it's also 1.625 and in fact you didn't have
to go down you can just go from this point and then go to the curve like this then you
pretty much will get to the next value and I will tell you this is approximately 1.76
and of course you can repeat the process so let me just zoom in right here now go to the line
then go to the curve go to the line go to the curve go to a line install as you can see this
sequence will be approaching too really cool huh and of course if you guys would like I will tell
you guys that the next value F of f of f of x this right here will be about 1.866 and then
you will get the next one it's about 1.93 and so on so on so they will be approaching two
really cool now let me also show you if you pick another number that say right here instead right
2.5 whatever again go to the Curve all right and then let me zoom in go to the curve go to
the line go to the curve go to the line go to the curve go to the line and so on so on
this sequence will also be approaching two and of course when we have all the sequence being
approaching 2 then of course they also found it right very cool and if you happen to pick this and
that because they are the fixed point then they are always just going to be that number right
and finally here's the official solution let me know what you guys think and as always that's it