Please look into the following question. I have solved Part (a). For part (b) I found $f_{\pi_{2}}$ and $f_{\pi_{3}}$, where $f_{\pi_{2}}$ stands for the partial of $f$ with respect to $\pi_{2}$. I found that there are 4 critical points $(1/2,1/2); (1/2,0); (0,1/2); (1/3,1/3)$. Here, $f=\pi_{1}^{2}$.
Among them first 3 cannot happen, otherwise it will cause $0$ in the denominator of $F$. So, the only option is $(1/3,1/3)$, which makes the triangle equilateral.
I am stuck with the last part of (b).." Explain why .......is a triangular region". and Part (c).
How do I approach these 2 problems? Please, any hints will be appreciated. For part (c), will it be ok, if I calculate using basic geometry the area of equilateral triangle and the corresponding circumscribed circle and then proceed to calculate $\pi_{1}$ for 2 different cases, thereby establishing the inequality mentioned in the question at the end?
Problem:
HINT
For point b note that $\Pi_2$ and $\Pi_3$ have to satisfy the following inequalities
$$\left(\frac12-\Pi_2\right)\left(\frac12-\Pi_3\right)\left(\Pi_2+\Pi_3-\frac12\right)\ge0$$
$$0\leq\Pi_2\le1\quad 0\leq\Pi_3\le1 \quad \frac12\leq\Pi_2+\Pi_3\le1$$