We now minimize $ \frac{1}{a} + \frac{1}{b} $ under the constraint $ a + b = 1 $. Using the method of Lagrange multipliers or AM-HM inequality:

["Title: Minimizing $ \frac{1}{a} + \frac{1}{b} $ Under Constraint $ a + b = 1 $: A Case Study Using AM-HM Inequality", "---", "Introduction", "Optimization is a cornerstone of applied mathematics, economics, and engineering. One classic problem—finding the minimum of $ \frac{1}{a} + \frac{1}{b} $ under the constraint $ a + b = 1 $, with $ a, b > 0 $—is a rich example that beautifully illustrates foundational optimization techniques. In this article, we explore this minimization problem using the powerful and intuitive AM-HM (Arithmetic Mean-Harmonic Mean) inequality, which provides a direct and elegant solution. We also briefly touch on the Lagrange multiplier method as a complementary analytical approach.", "This problem arises in contexts such as resource allocation, economics, and physics, where minimizing reciprocals under a fixed sum constraint often models efficiency, cost, or opportunity cost.", "---", "### Problem Statement", "Minimize\n$$\nf(a, b) = \frac{1}{a} + \frac{1}{b}\n$$\nsubject to the constraint\n$$\na + b = 1,\quad a > 0,, b > 0.\n$$", "---", "### Step 1: Use the AM-HM Inequality", "The AM-HM inequality states that for any positive real numbers $ a $ and $ b $:", "$$\n\frac{a + b}{2} \geq \frac{2}{\frac{1}{a} + \frac{1}{b}}.\n$$", "Equality holds if and only if $ a = b $.", "From the constraint $ a + b = 1 $, substitute into the inequality:", "$$\n\frac{1}{2} \geq \frac{2}{\frac{1}{a} + \frac{1}{b}}.\n$$", "Inverting both sides (and reversing the inequality):", "$$\n\frac{1}{a} + \frac{1}{b} \geq 4.\n$$", "Equality occurs when $ a = b $. Given $ a + b = 1 $, this implies $ a = b = \frac{1}{2} $.", "Thus, the minimum value of $ \frac{1}{a} + \frac{1}{b} $ is $ 4 $, achieved when $ a = b = \frac{1}{2} $.", "---", "### Step 2: Verification via Lagrange Multipliers", "For completeness, we validate the result using Lagrange multipliers—a method from calculus that solves constrained optimization problems.", "Define the Lagrangian:\n$$\n\mathcal{L}(a, b, \lambda) = \frac{1}{a} + \frac{1}{b} + \lambda(1 - a - b).\n$$", "Take partial derivatives and set them to zero:", "$$\n\frac{\partial \mathcal{L}}{\partial a} = -\frac{1}{a^2} - \lambda = 0 \quad \Rightarrow \quad \lambda = -\frac{1}{a^2},\n$$\n$$\n\frac{\partial \mathcal{L}}{\partial b} = -\frac{1}{b^2} - \lambda = 0 \quad \Rightarrow \quad \lambda = -\frac{1}{b^2},\n$$\n$$\n\frac{\partial \mathcal{L}}{\partial \lambda} = 1 - a - b = 0.\n$$", "From the first two equations:\n$$\n\frac{1}{a^2} = \frac{1}{b^2} \quad \Rightarrow \quad a = b \quad (\ ext{since } a, b > 0).\n$$", "Substitute into the constraint: $ a + b = 1 \Rightarrow 2a = 1 \Rightarrow a = \frac{1}{2},\ b = \frac{1}{2} $.", "Then,\n$$\n\frac{1}{a} + \frac{1}{b} = 2 + 2 = 4.\n$$", "Thus, both methods confirm the minimum is $ \boxed{4} $, achieved at $ a = b = \frac{1}{2} $.", "---", "### Step 3: Interpretation and Applications", "Why does $ \frac{1}{a} + \frac{1}{b} $ attain its minimum when $ a = b $? Intuitively:", "- When one argument becomes very small, its reciprocal becomes large, increasing the sum.\n- Symmetry suggests balance minimizes dispersion—this aligns with AM-HM: higher individual values yield smaller reciprocals.", "This principle applies widely:\n- In economics: minimizing unit-cost sum under total quantity constraint.\n- In physics: minimizing resistance or maximizing efficiency in parallel systems.\n- In engineering: optimizing material distribution under fixed resources.", "---", "### Conclusion", "The problem of minimizing $ \frac{1}{a} + \frac{1}{b} $ with $ a + b = 1 $ exemplifies how inequality principles and calculus-based methods converge to yield optimal solutions efficiently.", "Using the AM-HM inequality offers a rapid, insightful route, while Lagrange multipliers deliver rigorous verification. Together, they enrich our toolkit for tackling constrained optimization—handy both in theory and practice.", "Whether applied in research, finance, or system design, understanding such minimization fosters smarter decision-making grounded in mathematical elegance.", "---", "Keywords: minimization, $ \frac{1}{a} + \frac{1}{b} $, constraint optimization, AM-HM inequality, Lagrange multipliers, $ a + b = 1 $, $ a, b > 0 $, optimization techniques, inequality proofs, calculus, economic applications, mathematical tools.", "---", "Meta Description:\nLearn how to minimize $ \frac{1}{a} + \frac{1}{b} $ with $ a + b = 1 $ using AM-HM inequality and Lagrange multipliers—ideal for math students, data scientists, and engineers applying optimization in real-world problems."]