Lagrange Multiplier problem: max/min of f(x,y,z), constraint g(x,y,z)=k

[FoundBread]

Hi I'm having trouble with this Lagrange Multiplier problem. I've done a few other problems no problem but this one I can't wrap my head around at all.

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(b) The equation of a general ellipsoid centred at the origin is

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

for \(\displaystyle a,\, b,\, c\, \in\, (0,\, \infty).\) Its volume is \(\displaystyle \frac{4}{3}\pi abc.\) Let \(\displaystyle L\) be a fixed positive constant.
Given the constraint \(\displaystyle a\, +\, b\, +\, c\, =\, L,\) show that the ellipsoid with greatest volume is a sphere.

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I have the method:
f(x,y,z) is the function trying to get the max/min of it
g(x,y,z)=k is my constraint

and then I have f(x,y,z) - λ(g(x,y,z)-k)
and get the derivatives with relation to x,y,z,λ (presume it will be a,b,c,λ in my case)
Then let all them = 0 and sub my solutions into f.
But I really just am getting so stuck I'm not sure how to show a sphere is the best shape for it.
Any help would be greatly appreciated even just tips not looking for someone to do it for me. (It's from a former exam paper I'm studying for repeats)
Thanks

 

[FoundBread]

I've got the derivatives in relation to a,b,c,
And I've got
Fa => (-2x^2)/(a^3)= λ
Fb => (-2y^2)/(b^3)= λ
Fc => (-2z^2)/(c^3)= λ
Fλ => a + b + c = L

I'm struggling with this part not sure how to sort out this system of equations and get it back into the original function

 

[MarkFL]

Your objective function is the volume of the ellipsoid:

\(\displaystyle \displaystyle f(a,b,c)=\frac{4}{3}\pi abc\)

Subject to the constraint:

\(\displaystyle g(a,b,c)=a+b+c-L=0\)

And so Lagrange gives us:

\(\displaystyle \displaystyle \frac{4}{3}\pi bc=\lambda\)

\(\displaystyle \displaystyle \frac{4}{3}\pi ac=\lambda\)

\(\displaystyle \displaystyle \frac{4}{3}\pi ab=\lambda\)

What does this system imply?

 

[FoundBread]

\(\displaystyle \displaystyle \frac{4}{3}\pi bc=\lambda\)

\(\displaystyle \displaystyle \frac{4}{3}\pi ac=\lambda\)

\(\displaystyle \displaystyle \frac{4}{3}\pi ab=\lambda\)

What does this system imply?

So can I just say that a=b=c so it must be a sphere because the formula for the vol is 4/3\pi r^3?


(Thanks so much can't believe I spent so long using the wrong thing for my function)