I am trying to solve this equation:
Find max and min values of
$$ f(a,b,c) \left (\frac{1}{abc}\right ) $$
Subject to: $$ a^2 + b^2 + c^2 = 1\ $$
I calculated the system of equations
$$ \frac{1}{a^2bc} = 18\lambda a \ $$ $$ \frac{1}{ab^2c} = 2\lambda b \ $$ $$ \frac{1}{abc^2} = 2\lambda c \ $$
However from here I am not sure how I should proceed. Any advice on solving Lagrange multipliers in general and tips on the above equation would be much appreciated.
From
$$ \cases{ \frac{1}{a^2bc} = 18\lambda a \\ \frac{1}{ab^2c} = 2\lambda b \\ \frac{1}{abc^2} = 2\lambda c }\Rightarrow\cases{ \frac{1}{abc} = 18\lambda a^2 \\ \frac{1}{abc} = 2\lambda b^2 \\ \frac{1}{abc} = 2\lambda c^2 }\Rightarrow \frac{3}{abc} = 2\lambda $$
and now
$$ \cases{ \frac{1}{a^2bc} = 18\lambda a = 18a\frac 32\frac{1}{abc}\\ \frac{1}{ab^2c} = 2\lambda b =2b\frac 32\frac{1}{abc}\\ \frac{1}{abc^2} = 2\lambda c = 2c\frac 32\frac{1}{abc} }\Rightarrow\cases{a^2 = \frac{1}{27}\\ b^2 = \frac 13\\ c^2=\frac 13} $$