Hey so I need help with these



For 4 I get that 1 = -lambda(a^-2) and that ax^-2 = by^-2 = cz^-2 and that x + y + z = -lambda and a whole bunch of other equations but I cannot move on from there, just stuck with too many unknowns and the constraint doesn't really help much
For 5 I end up just getting cases where x = y = z = 0 or lambda = 0 but I do not think these are right...

For question 4 you are on the right track in saying

-λa/x^2 = -λb/y^2 = -λc/z^2

in fact, in this case the function λ disappears leaving you with only

x^2/a = y^2/b = z^2/c

now you can solve for y,z in terms of x, then plug it into your constraint
you get something like a/x + sqrt(ab)/x + sqrt(ac)/x = 1
then solve for x, and do the same for y,z


For problem 5

Let g(x,y,z) = x+y+2z=0 and h(x,y,z) = x^2/25 + y^2/25 + z^2/9 =1

Apply Lagrange Multipliers.

∇f = λ∇g + μ∇h
(2x,2y,2z> = λ<1,1,2> + μ<2x/25, 2y/25, 2z/9>

you have the following set of equations
2x = λ + μ2x/25
2y = λ + μ2y/25
2z = 2λ + μ2z/9
x+y+2z=0
x^2/25 + y^2/25 + z^2/9 =1

then you solve for x,y and z

I dont have time right now, but i might write out the steps later

For number 4 should I expect things to cancel out nicely or will I get an answer in terms of a,b,c?

And

2x = λ + μ2x/25
2y = λ + μ2y/25
2z = 2λ + μ2z/9
x+y+2z=0
x^2/25 + y^2/25 + z^2/9 =1

I have gotten to this point for 5 but I just end up with x = y = z = 0

Thank you for the help!

Yes your points will be in terms of a,b,c.


This post was edited by J0nn on Nov 4 2014 11:23pm

Alright, for question 5 you have

2x = λ + μ2x/25 (1)
2y = λ + μ2y/25 (2)
2z = 2λ + μ2z/9 (3)
x+y+2z=0 (4)
x^2/25 + y^2/25 + z^2/9 =1 (5)

first subtract (2) from (1)

2(x-y) = 2μ/25 (x - y)
you can see that either x = y or μ = 25
If x = y then

x + y + 2z = 0 ----> x = y = -z
and
x^2/25 + x^2/25 + x^2/9 = 1 -----> x = +/- 15/sqrt(43)

let S be the value of f(x,y,z)
S = 3(225/43) = 15.7

x≠y ---->μ=25 ----->2x=λ+2x ------>λ=0 ----->2z = 0 + 50z/9 ------> z=0.
So x=−y, and
x^2/25 + x^2/25 + 0 = 1 -----> 2x^2 = 25

S=x^2 + y^2 + z^2 = 2x^2 + 0 = 25. Thus S_min = 15.7, and S_max = 25

Just to let you check your result, answer for question 4 is :

a + b + c + 2*(sqrt(ab)+sqrt(ac)+sqrt(bc))

Feel free to pm if your calculations don't cancel out properly.

Oh okay that makes more sense I was dividing out the x-y term and was having issues when x=y

And yeah that is what I got for question 4
Thank you for all the help!!