Given x,y are real numbers that satisfy x^2 + 5y^2 - 5xy + 2y + 3 = 0
#Find the minimum value of P = xy + 2021
rose basin
tidal impBOT
Given x,y are real numbers that satisfy x^2 + 5y^2 - 5xy + 2y + 3 = 0
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copper viper
Given x,y are real numbers that satisfy x^2 + 5y^2 - 5xy + 2y + 3 = 0
looks like a simons favourite factoring trick
inland mason
So what have you tried so far
inland mason
hey so how can we proceed?
Since this seems to be pretty annoying to factor
copper viper
oh it is not sfft my bad
hmmm, try expressing one variable in terms of the other
then getting a function for P
rose basin
So what have you tried so far
stayed up to mid night and i made no progress whatsoever
any ideas here guys?
inland mason
I found a random pair that worked after logic bashing
This problem is too good for me lol
inland mason
stayed up to mid night and i made no progress whatsoever
It won't really help you though
inland mason
any ideas here guys?
Honestly I'm just setting boundaries and trying to plug
rose basin
hey so any solution?
copper viper
what does happen if you isolate a variable and sub it in
dry glade
Given x,y are real numbers that satisfy x^2 + 5y^2 - 5xy + 2y + 3 = 0
Try bringing the constraint to the canonical form first.
Well, maybe no need for the actual canonical form. Just complete the squares.
dry glade
No, that's not how you do it.
We have x^2 + 5xy. What else is needed for a perfect square?
Oh, actually, I have a better idea.
From the constraint:
xy = (1/5)x^2 + y^2 + (2/5)y + 3/5
So:
P = (1/5)x^2 + y^2 + (2/5)y + 3/5 + 2021
And now we complete the squares.
This is an elliptic paraboloid, so its minimum value is easy to find.
safe grail
differentiate it
partially
wrt x
you'll get x= 5y/2
for maximum value of this thing
then substitute this x and get a quadratic for y
dry glade
Or that, yeah.
Though, you can also just complete the square. Then you can do it without differentiation.
safe grail
alternatively, you could differentiate it wrt y too
this equation seems annoying for completing a square
dry glade
alternatively, you could differentiate it wrt y too
You need to find both partial derivatives, of course. This is a function of two variables, after all.
dry glade
this equation seems annoying for completing a square
Eh, it's not that bad.
Besides, there's just one square to complete.
safe grail
lemme see
safe grail
xy >= 8/5 ?
safe grail
any middle school solution?
completing the square is middle school 💀
dry glade
any middle school solution?
Completing the square. Though, this isn't a middle school level problem.
rose basin
but it was on my exam the other day
safe grail
this is not a middle school question at all
safe grail
completing the square is your only go then
lemme try it, though differentiation is the fastest solution
rose basin
ok ill wait
dry glade
That's not the correct way to do it.
dry glade
From the constraint:
xy = (1/5)x^2 + y^2 + (2/5)y + 3/5
So:
P = (1/5)x^2 + y^2 +...
And it's easier to use this approach.
dry glade
differentiation is my go to solution
Well, true, that will work, but that is a university level method.
dry glade
its an 11th grade solution
You don't learn differential analysis of functions of several variables in school.
safe grail
in our country, we do-
dry glade
Huh, really? That's very unusual.
safe grail
idk about other schools, but my school did teach it
immediately after 12th grade we give an exam called JEE
and the maths in their is just- brain wrecking
dry glade
Hm. Well, ok.
In that case, there are two approaches:
- Use Lagrange multipliers or bordered hessian.
- Get rid of the constraint by expressing xy, then just do the usual optimization.
stark imp
i tried lagrange multipliers and various other methods and they all result in quartic equations of a variable
dry glade
I think getting rid of the constraint and completing the square is the simplest elementary approach here.
The second easiest is getting rid of the constraint, then using the usual optimization approach.
rose basin
I think getting rid of the constraint and completing the square is the simplest ...
can you show me the full solution pls?
dry glade
can you show me the full solution pls?
Try it yourself first.
rose basin
ok but what is "getting rid of the constraint"?
dry glade
ok but what is "getting rid of the constraint"?
As I did above: you can express xy from it and substitute it into P.
rose basin
i give up
safe grail
😭 😭 😭 😭
rose basin
From the constraint:
xy = (1/5)x^2 + y^2 + (2/5)y + 3/5
So:
P = (1/5)x^2 + y^2 +...
i cant really see how we can progress forward here
can u show me?
dry glade
i cant really see how we can progress forward here
Complete the square with y. Then the result should be obvious.
rose basin
ah ok
i see
tks for the help
now how do i close this or it automatically closes itself?
dry glade
now how do i close this or it automatically closes itself?
You're welcome!
You can use +close to close the post.