Math Is Fun Forum

123ronnie321

Member

Registered: 2010-09-28

Posts: 128

Inequality

x,y,z>0 and are real and satisfy x + y + z = 1

Find minimum value of {(1/x)+1}{(1/y)+1}{(1/z)+1}.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Inequality

Hi 123ronnie321;

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

123ronnie321

Member

Registered: 2010-09-28

Posts: 128

Re: Inequality

I know that but how did you get the answer?

Yes, you have read the question right.

Last edited by 123ronnie321 (2010-12-04 23:23:03)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Inequality

I am sure that this can be worked with inequalities and maybe just with algebra. I always use the method of Lagrangian Multipliers.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

123ronnie321

Member

Registered: 2010-09-28

Posts: 128

Re: Inequality

You are right.
I did it using am gm inequality. Thanks for your time.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Inequality

I think that is the preferred way. I just like using a bit of vectors that the Lagrangian Multipliers method uses.

I remember having this particular discussion ( calculus vs. AM GM ) with another member here. She is an ace with inequalities so that is what she would probably use. I tend to cling to this principle:

It clearly means that opinions vary from person to person because I am the one who thinks that lagrangian multipliers is one of the very few things which can get you the correct answer even if you have no idea of how it works.
On the contrary applying inequalities like AM,GM require you to be very very careful because sometimes it has to be applied in a very subtle way.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

123ronnie321

Member

Registered: 2010-09-28

Posts: 128

Re: Inequality

I have never heard of lagrangian multipliers. Can you post a link where i can learn the basics of it?

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Inequality

Hi 123ronnie321;

If you have a good calculus book like Stewart or any other they will surely cover it. Or go here and choose the vid you understand.

http://www.youtube.com/results?search_q … ultipliers

If you have covered partial differentiation, some vectors ( optional ) and can solve a simultaneous set of equations ( the hard part ) you will be okay.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

123ronnie321

Member

Registered: 2010-09-28

Posts: 128

Re: Inequality

I saw some videos of lagrange multipliers. Thank you.
Can you give me a simple problem?

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Inequality

Hi 123ronnie321;

That is a little tough for me. To me all of them are hard. Try this one.

Maximize x+2y
Subject to the constraint x^2 + y^2 = 1

( If you have covered this one in the videos let me know )

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

123ronnie321

Member

Registered: 2010-09-28

Posts: 128

Re: Inequality

max x+2y = sqrrt(5) at x = 1/[sqrrt(5)], y = 2/[sqrrt(5)].
i saw a similar sum in the video but the constraint was linear i.e. opposite of this problem.
Graphically, if we make the line x+2y=c tangent to the circle x^2 + y^2 = 1, we get the same result.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Inequality

Hi;

That is correct!

http://www.ualberta.ca/MATH/gauss/fcm/c … _exmpl.htm

Maximize 3x - y - 3z Subject to:

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Inequality

I hope this is correct.

By AM–GM,

and

Hence

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Inequality

Hi;

I hope this is correct.

Hmmm, I think you already know the answer to that.

About the proof, nice to watch you work.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

123ronnie321

Member

Registered: 2010-09-28

Posts: 128

Re: Inequality

Hi bobby,
max = 2*sqrrt(6), min = -2*sqrrt(6).

Hi Jane,
Thank you.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Inequality

Hi 123ronnie321;

Very good! What do you think of the technique?

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

123ronnie321

Member

Registered: 2010-09-28

Posts: 128

Re: Inequality

Good! But it can get a bit lengthy sometimes. Still it is reliable.

sameer mishra

Member

Registered: 2010-08-27

Posts: 64

Re: Inequality

in such type of problem of inequality i.e
A.M>=G.M
maxima $ minima exist at point where equality will hold i.e "if x=y=z"
you can assume x=y=z=k;
given that x+y+z=1
hence 3k=1
k=1/3
put the valu $obtain the answer
my friend this trick is quite comfortable if you are preparing for "I.I.T"

sameer mishra

Member

Registered: 2010-08-27

Posts: 64

Re: Inequality

why all of you are going so large my friend
x^2+y^2=1
let x=sin(a);
y=cos(a);
x+2y=sin(a)+2cos(a)=squt(5)sin(a+t);
hence maximum valu=squt(5), where t=tan^-1(2)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Inequality

No one is going large or small. The idea was to find the maximum or minimum by the Method of Lagrangian Multipliers. It was an exercise to become familiar with that technique. Using another idea sort of defeats the purpose of the exercise.

Also you should check your work because you have the wrong answer.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

sameer mishra

Member

Registered: 2010-08-27

Posts: 64

Re: Inequality

x+y-z=0  ;
z=x+y;
Max(3x-y-3z);
put the valu of z
Max(-4y)=.........(d)
x^2+ 2y^2=1............(A)
lets assume trigonometric variable which setisfied (A)
x=sin(k),z=cos(t)/squt(2);
y=[cos(t)/squt(2)  -sin(t)];
From (d)
Max(-4y)=max (4sin(t)-2*squt(2)cos(t));
solve like previous question
obtain squt(24),
"TELL ME FRIEND about process ans may be wrong due to calculation mistake"

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Inequality

That answer is also not √5 which is what you originally put.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

sameer mishra

Member

Registered: 2010-08-27

Posts: 64

Re: Inequality

sorry at place k put t

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Inequality

Okay, besides from that, the question was to give 123ronnie123 a couple of problems in the Method of Lagrangian Multipliers. They are for illustration and practice in that method. You can solve them any way you like but you cannot say he used the wrong method. He had no choice.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

sameer mishra

Member

Registered: 2010-08-27

Posts: 64

Re: Inequality

friend bobby ask good question of maxima $minima
tell me about true answer