# Lagrange multipliers

1. Sep 25, 2005

### thenewbosco

Find the shortest and longest distance from the origin to the curve
$$x^2 + xy + y^2=16$$ and give a geometric interpretation...the hint given is to find the maximum of $$x^2+y^2$$

i am not sure what to do for this problem

thanks

2. Sep 25, 2005

### amcavoy

Are you sure you need Lagrange Multipliers for this?

3. Sep 25, 2005

### thenewbosco

it says for the hint to use the method of lagrange multipliers to find the maximum of $$x^2 + y^2$$ but i am not sure how to do it using any method, so any help is appreciated.

4. Sep 25, 2005

### leon1127

Solve for y. use rate of change respect to the distance.
that is the "cal 1 method"

the path equation is constraint i think. apply Lagrange Multipliers on the distance formula

Last edited: Sep 25, 2005
5. Sep 25, 2005

### thenewbosco

solve for y in what though. in the question it says $$x^2+y^2$$ this isnt even an equation though.
im sorry i still dont get it

6. Sep 25, 2005

### leon1127

you can solve for y in tern of x
and then using the distance formula D = (y^2+x^2)^0.5
sub the y equation into the distance formula
take the first derivative
fine 0s
test it
done

that is cal 1 method, it requires a lot of work

$$x^2+y^2$$ looks really similar to the distance formula
$$D^2 = x^2 + y^2$$

you can set $$D = f(x)$$ or $$D^2 = f(x)$$ and find the del of it, since the square doesnt where the extreme occurs, therefore the text tells you to fine the max of $$x^2+y^2$$

Last edited: Sep 25, 2005