Question: An entrepreneur designing a sustainable irrigation system models soil moisture retention with a function $ f $ such that $ f(x + y) + f(x - y) = 2f(x) + 2f(y) $ for all real $ x, y $, and $ f(1) = 3 $. Find $ f(5) $.
["Optimizing Water Use in Agriculture: Solving the Irrigation Model with a Functional Equation", "In sustainable farming, precise water management is essential. One innovative approach involves designing efficient irrigation systems modeled using mathematical functions that capture soil moisture retention dynamics. A recent breakthrough by an entrepreneur in agritech models these retention patterns with a functional equation, leading to a powerful method for predicting water distribution.", "### The Functional Equation", "The entrepreneur establishes a function $ f: \mathbb{R} \ o \mathbb{R} $ that satisfies the identity:\n$$\nf(x + y) + f(x - y) = 2f(x) + 2f(y)\n$$\nfor all real numbers $ x $ and $ y $, and is given that $ f(1) = 3 $. This type of functional equation is well-known in mathematical analysis and has a distinctive general solution under mild regularity conditions (e.g., continuity or measurability, often assumed implicitly in applied problems).", "This equation is a form of Jensen’s functional equation generalized over two variables. Its solutions over $ \mathbb{R} $ are quadratic functions when regularity conditions hold. Specifically, the general solution is:\n$$\nf(x) = ax^2\n$$\nfor some constant $ a \in \mathbb{R} $.", "### Verifying the Solution and Finding $ a $", "Assume $ f(x) = ax^2 $. Substitute into the functional equation:\n$$\nf(x + y) + f(x - y) = a(x + y)^2 + a(x - y)^2 = a(x^2 + 2xy + y^2 + x^2 - 2xy + y^2) = a(2x^2 + 2y^2) = 2ax^2 + 2ay^2\n$$\nOn the other hand:\n$$\n2f(x) + 2f(y) = 2ax^2 + 2ay^2\n$$\nBoth sides match, confirming $ f(x) = ax^2 $ satisfies the equation for any real $ a $.", "Now use the given condition $ f(1) = 3 $:\n$$\nf(1) = a(1)^2 = a = 3 \quad \Rightarrow \quad a = 3\n$$\nThus, the function is $ f(x) = 3x^2 $.", "### Calculating $ f(5) $", "$$\nf(5) = 3 \cdot (5)^2 = 3 \cdot 25 = 75\n$$", "### Practical Implications for Sustainable Irrigation", "This mathematical insight enables the entrepreneur to design optimal irrigation layouts. By modeling soil moisture retention precisely through such functional relationships, water delivery systems can be calibrated to maintain ideal soil hydration—minimizing waste while maximizing crop yield. The quadratic nature of $ f(x) $ reflects how moisture retention often scales with distance from a water source or time, aiding in predictive modeling.", "In summary, solving this functional equation not only yields a clean mathematical form but also empowers real-world applications in sustainable agriculture through accurate, data-driven irrigation design.", "Final answer: $ f(5) = \boxed{75} $"]
