Find maxima and minima for U(x,y,z) = x^2+y^2+z^2 with constraint of ellipsoid

[edward]

Hello , I got really messy with trying to solving the next Lagrange multiplier problem:
Find maxima and minima for:

\(\displaystyle U(x,\, y,\, z)\, =\, x^2\, +\, y^2\, +\, z^2\)

with the constraint of ellipsoid:

\(\displaystyle \dfrac{x^2}{a^2}\, +\, \dfrac{y^2}{b^2}\, +\, \dfrac{z^2}{c^2}\, =\, 1\)

managed to find 6 critical points:

P₁: (a, 0, 0, -a²)
P₂: (-a, 0, 0, -a²)
P₃: (0, b, 0, -b²)
P₄: (0, -b, 0, -b²)
P₅: (0, 0, c, -c²)
P₆: (0, 0, -c, -c²)

and couldn't progress any further.

Thanks!

 

[Deleted member 4993]

Hello , I got really messy with trying to solving the next Lagrange multiplier problem:
Find maxima and minima for:

with the constraint of ellipsoid:

managed to find 6 critical points P1(a,0,0,-a^2),P2(-a,0,0,-a^2),P3(0,b,0,-b^2),P4(0,-b,0,-b^2),P5(0,0,c,-c^2),P6(0,0,-c,-c^2)
and couldn't progress any further.

Thanks!

How did you calculate the critical points?

Please share work.

 

[edward]

How did you calculate the critical points?

Please share work.

\(\displaystyle L(x,\, y,\,z,\, \lambda)\, =\, x^2\, +\,y^2\,+\, z^2\, +\,\lambda\left(\dfrac{x^2}{a^2}\, +\,\dfrac{y^2}{b^2}\, +\, \dfrac{z^2}{c^2}\, -\,1\right)\)

than calculated the grad(L) = (0, 0, 0, 0)

from the 4 equations I extracted the 6 critical points.

Than I calculated the hessian matrix for each point and every hessian equal to zero , from here I have no idea how to proceed.