Question: Let $ x, y, z $ be positive real numbers such that $ x + y + z = 1 $. Find the minimum value of $ \frac{1}{x} + \frac{4}{y} + \frac{9}{z} $.

["Optimizing with Constraints: Finding the Minimum of $ \frac{1}{x} + \frac{4}{y} + \frac{9}{z} $ Given $ x + y + z = 1 $", "When working with optimization problems under constraints, the method of Lagrange multipliers or application of inequality principles like Cauchy-Schwarz or Jensen’s inequality can yield elegant solutions. Here, we are given that $ x, y, z > 0 $ and $ x + y + z = 1 $, and we aim to minimize the expression:", "$$\nS = \frac{1}{x} + \frac{4}{y} + \frac{9}{z}\n$$", "### Applying the Cauchy-Schwarz Inequality", "We apply the Cauchy-Schwarz Inequality in the Engel form (also known as Titu’s Lemma):", "$$\n\left( \frac{a_1^2}{b_1} + \frac{a_2^2}{b_2} + \frac{a_3^2}{b_3} \right) \geq \frac{(a_1 + a_2 + a_3)^2}{b_1 + b_2 + b_3}\n$$", "We aim to match the numerator $ 1, 4, 9 $ to $ a_1 = 1, a_2 = 2, a_3 = 3 $. Set:", "- $ a_1 = 1 \Rightarrow \frac{1^2}{x} $\n- $ a_2 = 2 \Rightarrow \frac{2^2}{y} = \frac{4}{y} $\n- $ a_3 = 3 \Rightarrow \frac{3^2}{z} = \frac{9}{z} $", "Apply Titu’s Lemma:", "$$\n\frac{1^2}{x} + \frac{2^2}{y} + \frac{3^2}{z} \geq \frac{(1 + 2 + 3)^2}{x + y + z} = \frac{6^2}{1} = 36\n$$", "Thus,", "$$\n\frac{1}{x} + \frac{4}{y} + \frac{9}{z} \geq 36\n$$", "### Equality Condition", "Equality in Cauchy-Schwarz holds when:", "$$\n\frac{a_1}{\sqrt{b_1}} = \frac{a_2}{\sqrt{b_2}} = \frac{a_3}{\sqrt{b_3}\n\quad \Rightarrow \quad\n\frac{1}{\sqrt{x}} = \frac{2}{\sqrt{y}} = \frac{3}{\sqrt{z}} = k \quad \ ext{(for some } k > 0 \ ext{)}\n$$", "From this, express $ x, y, z $ in terms of $ k $:", "- $ \sqrt{x} = \frac{1}{k} \Rightarrow x = \frac{1}{k^2} $\n- $ \sqrt{y} = \frac{2}{k} \Rightarrow y = \frac{4}{k^2} $\n- $ \sqrt{z} = \frac{3}{k} \Rightarrow z = \frac{9}{k^2} $", "Now apply the constraint $ x + y + z = 1 $:", "$$\n\frac{1}{k^2} + \frac{4}{k^2} + \frac{9}{k^2} = \frac{14}{k^2} = 1 \Rightarrow k^2 = 14\n$$", "Then:", "- $ x = \frac{1}{14}, \quad y = \frac{4}{14} = \frac{2}{7}, \quad z = \frac{9}{14} $", "All values are positive and satisfy $ x + y + z = 1 $. Substituting into $ S $:", "$$\nS = \frac{1}{1/14} + \frac{4}{2/7} + \frac{9}{9/14} = 14 + 14 + 14 = 36\n$$", "Hence, the minimum value is achieved and equals $ 36 $.", "### Conclusion", "By using the Cauchy-Schwarz inequality and verifying equality conditions, we find that the minimum value of $ \frac{1}{x} + \frac{4}{y} + \frac{9}{z} $ under $ x + y + z = 1 $ is:", "$$\n\boxed{36}\n$$"]