<s>[INST] Find minimum of $a+b+c+\frac1a+\frac1b+\frac1c$ given that: $a+b+c\le \frac32$ 
Find minimum of $a+b+c+\frac1a+\frac1b+\frac1c$ given that: $a+b+c\le \frac32$ ($a,b,c$ are positive real numbers).

There is a solution, which relies on guessing the minimum case happening at $a=b=c=\frac12$ and then applying AM-GM inequality,but what if one CANNOT guess that?!
 [/INST] You may use Lagrange multipliers method to show that we need $a=b=c$. The optimisation problem would be to minimise $a+b+c+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$, having the constraint $a+b+c=t , t\leq \frac{3}{2}$.
$$F=a+b+c+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\lambda(a+b+c-t)$$
Then, we need to find the right $t$, for which the minimum is achieved. Knowing that $a=b=c$, the problem is simplified to minimizing
$$3(a+\frac{1}{a})$$
As mentioned in comments, it is a convex problem and it achieves its minimum at $a=1$. However, the constraint $a+b+c\leq\frac{3}{2}$ does not allow $a=1$. So, because of convexity, you should decrease $a$ until you satisfy the constraint.
</s>