Let f(a,b,c)=a2b2c2
Where a2+b2+c2=r2
⟹g(a,b,c)=a2+b2+c2
Use Lagrange multipliers to find the maximum value of a2b2c2
∇f(a,b,c)=λ∇g
⟹<2ab2c2,2a2bc2,2a2b2c>=λ<2a,2b,2c>
⟹ab2c2=aλ , a2bc2=bλ , a2b2c=cλ
⟹λ=b2c2 , λ=a2c2 , λ=a2b2
⟹b2c2=a2c2=a2b2⟹a=b=c
Substitute a=b=c in a2+b2+c2=r2
a=b=c=3r
(a2b2c2)max=(3r2)3
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