Question: Let $x, y, z$ be positive real numbers such that $2x + 3y + 4z = 12$. Find the minimum value of $x^2 + y^2 + z^2$.
["# Minimizing $x^2 + y^2 + z^2$ Subject to $2x + 3y + 4z = 12$: A Solution Using the Cauchy-Schwarz Inequality", "When given a constrained optimization problem involving a linear constraint and a quadratic objective, one of the most effective tools is the Cauchy-Schwarz inequality. In this article, we explore how to find the minimum value of $x^2 + y^2 + z^2$ for positive real numbers $x, y, z$ satisfying $2x + 3y + 4z = 12$.", "## The Problem Statement", "Let $x, y, z > 0$ be real numbers such that:\n$$\n2x + 3y + 4z = 12\n$$\nWe aim to minimize the expression:\n$$\nx^2 + y^2 + z^2\n$$", "## Why Cauchy-Schwarz?", "The constraint is a linear combination, and the objective is a sum of squares — a perfect scenario to apply the Cauchy-Schwarz inequality in $\mathbb{R}^3$:", "$$\n(a_1^2 + a_2^2 + a_3^2)(b_1^2 + b_2^2 + b_3^2) \geq (a_1b_1 + a_2b_2 + a_3b_3)^2\n$$", "Equality holds when $(a_1, a_2, a_3)$ and $(b_1, b_2, b_3)$ are proportional.", "## Applying Cauchy-Schwarz", "Choose constants $a_1 = 2$, $a_2 = 3$, $a_3 = 4$, and variables $b_1 = x$, $b_2 = y$, $b_3 = z$ — all positive. Then:", "$$\n(2^2 + 3^2 + 4^2)(x^2 + y^2 + z^2) \geq (2x + 3y + 4z)^2\n$$", "Compute the coefficients:\n$$\n2^2 + 3^2 + 4^2 = 4 + 9 + 16 = 29\n$$\n$$\n2x + 3y + 4z = 12 \Rightarrow (2x + 3y + 4z)^2 = 144\n$$", "So:", "$$\n29(x^2 + y^2 + z^2) \geq 144\n\Rightarrow x^2 + y^2 + z^2 \geq \frac{144}{29}\n$$", "## Achieving Equality", "Equality in Cauchy-Schwarz occurs when:\n$$\n\frac{x}{2} = \frac{y}{3} = \frac{z}{4} = k \quad \ ext{for some } k > 0\n$$", "Let $x = 2k$, $y = 3k$, $z = 4k$. Substitute into the constraint:", "$$\n2(2k) + 3(3k) + 4(4k) = 4k + 9k + 16k = 29k = 12\n\Rightarrow k = \frac{12}{29}\n$$", "Then:", "$$\nx = \frac{24}{29},\quad y = \frac{36}{29},\quad z = \frac{48}{29}\n$$", "Now compute $x^2 + y^2 + z^2$:", "$$\nx^2 + y^2 + z^2 = \left(\frac{24}{29}\right)^2 + \left(\frac{36}{29}\right)^2 + \left(\frac{48}{29}\right)^2 = \frac{576 + 1296 + 2304}{841} = \frac{4176}{841} = \frac{144}{29}\n$$", "This confirms the lower bound is achieved.", "## Conclusion", "The minimum value of $x^2 + y^2 + z^2$ under the constraint $2x + 3y + 4z = 12$ is $\frac{144}{29}$. This result is elegant, elegant, and efficient — showcasing the power of the Cauchy-Schwarz inequality in solving constrained optimization problems.", "Whether you're studying optimization, preparing for competitive exams, or applying geometry to real-world constraints, mastering such techniques is essential.", "---", "Keywords: minimum value, $x^2 + y^2 + z^2$, constraint optimization, Cauchy-Schwarz inequality, $2x + 3y + 4z = 12$, positive real numbers, inequality proof, $\mathbb{R}^3$, math optimization."]
