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Home » x^2 - y^2 = (x - y)(x + y) = 4 \cdot 10 = 40 - AIKO, infinite ways to autonomy.

x^2 - y^2 = (x - y)(x + y) = 4 \cdot 10 = 40 - AIKO, infinite ways to autonomy.

📅 May 2, 2026 👤 scraface

May 02, 2026

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📌 Evaluating \( R''(-2) \) shows it is positive, indicating a local minimum, so we consider behavior around \( x = -2 \).

📌 As \( x \to \pm \infty \), \( R(x) \) approaches \( \ln(x^2) \to \infty \), confirming \( x = -2 \) is a local maximum point in context.

📌 Thus, the value of \( x \) that maximizes \( R(x) \) in meaningful range is \(\boxed{-2}\).

📌 In an ecological study, the interaction between two species is modeled by the function \( h(x, y) = \frac{xy}{x^2 + y^2 + 1} \). Find the value of \( h(x, y) \) when \( x = y \) and \( x \to 1 \).

📌 Given \( h(x, y) = \frac{xy}{x^2 + y^2 + 1} \), evaluate at \( x = y \):

📌 \[ h(x, x) = \frac{x \cdot x}{x^2 + x^2 + 1} = \frac{x^2}{2x^2 + 1} \]

📌 Now assess the limit as \( x \to 1 \):

📌 \[ \lim_{x \to 1} h(x, x) = \frac{1^2}{2(1)^2 + 1} = \frac{1}{2 + 1} = \frac{1}{3} \]

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