Question 42
Let $M = \{(a,b,c) \in \mathbb{R}^3: 0 \lt a, b, c \lt \frac{1}{2} \text{ with } a+b+c=1\}$ and $f: M \to \mathbb{R}$ given as
$$
f(a, b, c) = 4\left(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right) - \frac{1}{abc}
$$
Find the best (real) bounds $\alpha$ and $\beta$ such that
$$
f(M) = \{f(a, b, c) : (a, b, c) \in M\} \subseteq [\alpha, \beta]
$$
and determine whether any of them is achievable.
$$
f(a, b, c) = 4\left(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right) - \frac{1}{abc}
$$
Find the best (real) bounds $\alpha$ and $\beta$ such that
$$
f(M) = \{f(a, b, c) : (a, b, c) \in M\} \subseteq [\alpha, \beta]
$$
and determine whether any of them is achievable.