<s>[INST] A question regarding minimum value of... $a+b+c=2$
$a,b,c \in \mathbb{R}^+$
What is The min value of $1/a+4/b+9/c$?
I have tried to use geometric shapes and aritmetic harmonic geometric means but unfortunately ı am going nowhere?
What do you suggest?
 [/INST] Lagrange multiplier
$$f(a,b,c,k)=\frac{1}{a}+\frac{4}{b}+\frac{9}{c}+k (a+b+c-2)$$
Solve the system $f'_x=0,f'_y=0,f'_z=0,f'_k=0$
That is
$$k-\frac{1}{a^2}=0,k-\frac{4}{b^2}=0,k-\frac{9}{c^2}=0,a+b+c-2=0$$
there are $3$ solutions but only one leads to the minimum and respects constraints
$a= \frac{1}{3},b=\frac{2}{3},c= 1$ for which $\frac{1}{a}+\frac{4}{b}+\frac{9}{c}=18$
Hope this helps
</s>