Lagrange Multipliers

L'prof

I'm stuck with this question and I want to figure out how to solve it in case a similar one comes up in my exam tomorrow!

Determine the max/min distance from the origin to any point on the ellipsoid defined by x^2 + (y^2)/9 + (z^2)/4 = 1

so f(x,y,z) = x^2 + y^2 + z^2 and g(x,y,z) = x^2 + (y^2)/9 + (z^2)/4 - 1

(2x, 2y, 2z) - λ(2x, 2y/9, z/2) = 0

but that gives me λ = 1, λ = 9, λ = 4 which is rubbish so I assume my approach is wrong for this question?

Thanks,
Jay

MathsManiac

jasonorr wrote: »

but that gives me λ = 1, λ = 9, λ = 4

No, it doesn't necessarily (at least not without combining it with a fourth equation. You've three equations in four unknowns there, giving infinitely many solutions. You must also include the partial diff wrt λ of the auxiliary function, or, equivalently, g(x,y,z)=0.
The solutions are then:

{x = 0, z = 2, λ = 4, y = 0}, {x = 0, z = -2, λ = 4, y = 0}, {x = 0, z = 0, λ = 9, y = 3}, {x = 0, z = 0, λ = 9, y = -3}, {z = 0, λ = 1, x = 1, y = 0}, {z = 0, λ = 1, x = -1, y = 0}

These solution sets give the local maima and minima. Substituting them into f gives your distances. The minimum distance is 1, corresponding to the two points above where λ=1; the max of f is 9 (giving a max dist of 3), at the two points where λ=9.

Those solutions should not surprise you, if you know how the shape of an ellipsoid in standard position corresponds to its equation.

Not certain how you got your λ-values, but they were right, provided you interpret them correctly and finish off. Why did you assert that they were rubbish?

If this is helpful, I hope you pick it up in the morning before your exam!

L'prof

MathsManiac wrote: »

Not certain how you got your λ-values, but they were right, provided you interpret them correctly and finish off. Why did you assert that they were rubbish?

If this is helpful, I hope you pick it up in the morning before your exam!

Hey, thanks for that! It's an afternoon exam so, I'll give that a go when I get in to college!

I just thought it was rubbish because I thought I was only supposed to get 1 value for λ!

I'll let you know if I can figure it out anyway and if a similar example comes up in the exam!

Thanks