<s>[INST] Find the minimum using lagrange I've got the formula $$x^3+y^3+z^3$$
With the constraint $ax+by+cz = 1$
I tried to solve this using lagrange but every possible way I try to use does not get me to the right answer
Using the lagrange I got to
$$3x^2+ka = 0 $$
$$3y^2+kb = 0 $$
$$3z^2+kc = 0 $$
=
$$3x^2bc+kabc = 0 $$
$$3y^2ac+kabc = 0 $$
$$3z^2ab+kabc = 0 $$
$$z = \sqrt{\frac{cy^2}{b}} $$
$$x = \sqrt{\frac{ay^2}{b}} $$
Fill this in the constraint and I then I cannot figure it out
Any got a clue how to solve this properly?
 [/INST] Assuming $x \ge 0, y \ge 0, z \ge 0$ to assure a bounded solution, we have from the stationary conditions
$$
x = \sqrt\frac{\lambda a}{3}\\
y = \sqrt\frac{\lambda b}{3}\\
z = \sqrt\frac{\lambda c}{3}\\
$$
and substituting into the restriction
$$
\frac{\lambda}{3} = \frac{1}{(a\sqrt a+b\sqrt b+c\sqrt c)^2}
$$
but
$$
x^3+y^3+z^3 = \left(\frac{\lambda}{3}\right)^{\frac 32}(a\sqrt a+b\sqrt b+c\sqrt c)
$$
hence
$$
\left\{\min_{x>0,y>0,z>0}(x^3+y^3+z^3)\ \ \mbox{s. t.}\ \ \ ax+by+cz=1\right\} = \frac{1}{(a\sqrt a+ b\sqrt b+c\sqrt c)^2}
$$
Now the bordered Hessian
$$
H_g = \left(
\begin{array}{cccc}
 0 & -a & -b & -c \\
 -a & 6 x & 0 & 0 \\
 -b & 0 & 6 y & 0 \\
 -c & 0 & 0 & 6 z \\
\end{array}
\right)
$$
has value 
$$
\det(H_g) = -36 \left(a^2 y z+b^2 x z+c^2 x y\right)
$$
Which with $x>0,y>0,z>0$ characterizes a minimum.
</s>