MHB GWR309's question at Yahoo Answers regarding Lagrange multipliers

MarkFL
Gold Member
MHB
Messages
13,284
Reaction score
12
Here is the question:

Please help with lagrange multipliers?

I don't understand how to do these at all. Here is one of the problems I have to do:

f(x,y) = xy x-2y=1Additional Details i thought I had more spaces when I typed... xy and x-2y=1 are 2 different equations.

Here is a link to the question:

Please help with lagrange multipliers? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.
 
Mathematics news on Phys.org
Hello GWR309,

We are given the objective function:

$$f(x,y)=xy$$

subject to the constraint:

$$g(x,y)=x-2y-1=0$$

Using Lagrange multipliers, we get the system:

$$y=\lambda$$

$$x=-2\lambda$$

and this implies:

$$x=-2y$$

Substituting for $x$ into the constraint, we find:

$$-2y-2y-1=0$$

$$y=-\frac{1}{4}\,\therefore\,x=\frac{1}{2}$$

And so the critical point is:

$$(x,y)=\left(\frac{1}{2},-\frac{1}{4} \right)$$

and the optimize value for the objective function is:

$$f\left(\frac{1}{2},-\frac{1}{4} \right)=-\frac{1}{8}$$

Since other test points such as $$\left(0,-\frac{1}{2} \right)$$ and $$\left(1,0 \right)$$ give $$f(x,y)>-\frac{1}{8}$$ we may conclude that:

$$f_{\min}=f\left(\frac{1}{2},-\frac{1}{4} \right)=-\frac{1}{8}$$

As a check, we may use the constraint to write $f$ as a function of 1 variable, namely:

$$f(x)=\frac{x}{2}(x-1)$$

We can see now that $f(x)$ is a parabola opening upwards, and we know its vertex (which is the global minimum) will line on the axis of symmetry, which is midway between the roots, on the line:

$$x=\frac{1}{2}$$

and so the minimum is:

$$f_{\min}=f\left(\frac{1}{2} \right)=-\frac{1}{8}$$

To GWR309 and any other guests viewing this topic, I invite and encourage you to post other optimization problems in our http://www.mathhelpboards.com/f10/ forum.

Best Regards,

Mark.
 
MarkFL said:
Hello GWR309,

We are given the objective function:

$$f(x,y)=xy$$

subject to the constraint:

$$g(x,y)=x-2y-1=0$$

Using Lagrange multipliers, we get the system:

$$y=\lambda$$

$$x=-2\lambda$$

and this implies:

$$x=-2y$$

Substituting for $x$ into the constraint, we find:

$$-2y-2y-1=0$$

$$y=-\frac{1}{4}\,\therefore\,x=\frac{1}{2}$$

And so the critical point is:

$$(x,y)=\left(\frac{1}{2},-\frac{1}{4} \right)$$

and the optimize value for the objective function is:

$$f\left(\frac{1}{2},-\frac{1}{4} \right)=-\frac{1}{8}$$

Since other test points such as $$\left(0,-\frac{1}{2} \right)$$ and $$\left(1,0 \right)$$ give $$f(x,y)>-\frac{1}{8}$$ we may conclude that:

$$f_{\min}=f\left(\frac{1}{2},-\frac{1}{4} \right)=-\frac{1}{8}$$

As a check, we may use the constraint to write $f$ as a function of 1 variable, namely:

$$f(x)=\frac{x}{2}(x-1)$$

We can see now that $f(x)$ is a parabola opening upwards, and we know its vertex (which is the global minimum) will line on the axis of symmetry, which is midway between the roots, on the line:

$$x=\frac{1}{2}$$

and so the minimum is:

$$f_{\min}=f\left(\frac{1}{2} \right)=-\frac{1}{8}$$

To GWR309 and any other guests viewing this topic, I invite and encourage you to post other optimization problems in our http://www.mathhelpboards.com/f10/ forum.

Best Regards,

Mark.

Where did you get the system though? this:
y=λ

x=−2λ
 
GWR309 said:
Where did you get the system though? this:
y=λ

x=−2λ

Hello GWR309,

Glad you joined us here! (Cool)

The method of Lagrange multipliers states:

To find the extrema of $z=f(x,y)$ subject to the constraint $g(x,y)=0$, solve the system of equations:

$$f_x(x,y)=\lambda g_x(x,y)$$

$$f_y(x,y)=\lambda g_y(x,y)$$

$$g(x,y)=0$$

Among the solutions $(x,y,\lambda)$ of the system will be the points $$\left(x_i,y_i \right)$$, where $f$ has an extremum. When $f$ has a maximum (or minimum), it will be the largest (or smallest) number in the list of functional values $$f\left(x_i,y_i \right)$$.

So, given that:

$$f(x,y)=xy$$

$$g(x,y)=x-2y-1=0$$

we then find by computing the first partials, that:

$$f_x(x,y)=y,\,f_y(x,y)=x,\,g_x(x,y)=1,\,g_y(x,y)=-2$$

and so the system we are to solve is:

$$y=\lambda$$

$$x=-2\lambda$$

$$g(x,y)=x-2y-1=0$$

From the first two equations, we find:

$$\lambda=y=-\frac{x}{2}\,\therefore\,x=-2y$$

and the rest follows as in my second post above.
 
for general lagrange multipliers problems, how do I determine if it is a max, min or neither?
 
Last edited:
GWR309 said:
for general lagrange multipliers problems, how do I determine if it is a max, min or neither?

You could check the Hessian matrix. Or check that your function is convex (which would imply you have a minimum). Or even just substitute in the value and compare it to the endpoints of your function...
 
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
I'm interested to know whether the equation $$1 = 2 - \frac{1}{2 - \frac{1}{2 - \cdots}}$$ is true or not. It can be shown easily that if the continued fraction converges, it cannot converge to anything else than 1. It seems that if the continued fraction converges, the convergence is very slow. The apparent slowness of the convergence makes it difficult to estimate the presence of true convergence numerically. At the moment I don't know whether this converges or not.
Back
Top