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Let $ s = \sin x \cos x $. Then:
f(x) = 1 + 2s + \frac{1}{s}
Note $ s = \frac{1}{2} \sin 2x $, and since $ 0 < x < \frac{\pi}{2} $, $ 0 < 2x < \pi $, so $ 0 < \sin 2x \leq 1 $, hence $ 0 < s \leq \frac{1}{2} $.
Define $ g(s) = 1 + 2s + \frac{1}{s} $, for $ 0 < s \leq \frac{1}{2} $.
g'(s) = 2 - \frac{1}{s^2}
Set $ g'(s) = 0 \Rightarrow 2 = \frac{1}{s^2} \Rightarrow s^2 = \frac{1}{2} \Rightarrow s = \frac{1}{\sqrt{2}} \approx 0.707 > \frac{1}{2} $, not in domain.
So minimum occurs at endpoint $ s = \frac{1}{2} $:
g\left( \frac{1}{2} \right) = 1 + 2 \cdot \frac{1}{2} + \frac{1}{\frac{1}{2}} = 1 + 1 + 2 = 4
Thus, the minimum value is $ \boxed{4} $.
Question: A philosopher of science considers three unit vectors $ \mathbf{a}, \mathbf{b}, \mathbf{c} $ in $ \mathbb{R}^3 $, representing distinct but balanced perspectives in a scientific model. Find the largest possible value of $ \|\mathbf{a} + \mathbf{b} + \mathbf{c}\| $.
