Question:

<div class="solution"> <p><strong>Step 1:</strong> To solve this problem, we first need to understand the given function \(f(x)\) and its definition for both \(x \leq 0\) and \(x > 0\). For \(x \leq 0\), \(f(x) = \max.\,(x^2, 1)\), which means \(f(x)\) will be the maximum value between \(x^2\) and \(1\). Since \(x^2\) is always positive and for \(x \leq 0\), \(x^2 \geq 1\) when \(x \leq -1\) and \(x^2 < 1\) when \(-1 < x \leq 0\), we can deduce that for \(x \leq -1\), \(f(x) = x^2\) and for \(-1 < x \leq 0\), \(f(x) = 1\).</p> <p><strong>Step 2:</strong> For \(x > 0\), \(f(x) = \min.\,(\{x\},\,|1 - |x||)\). Here, \(\{x\}\) denotes the fractional part of \(x\), and \(|1 - |x||\) is the absolute value of the difference between \(1\) and the absolute value of \(x\). To evaluate \(f(f(-5/2))\), we start by finding \(f(-5/2)\). Since \(-5/2 \leq 0\), we use the definition \(f(x) = \max.\,(x^2, 1)\). Thus, \(f(-5/2) = \max.\,((-5/2)^2, 1) = \max.\,(25/4, 1) = 25/4\).</p> <p><strong>Step 3:</strong> Now, we need to find \(f(25/4)\). Since \(25/4 > 0\), we use the definition \(f(x) = \min.\,(\{x\},\,|1 - |x||)\). The fractional part of \(25/4\) is \(1/4\), and \(|1 - |25/4|| = |1 - 25/4| = |-21/4| = 21/4\). Therefore, \(f(25/4) = \min.\,(1/4, 21/4) = 1/4\).</p> <p><strong>Answer:</strong> Based on the calculations, the correct answer is \(f(f(-5/2)) = 1/4\), which corresponds to option (c).</p> <div class="key-concept"><strong>Key Concept:</strong> Understanding the piecewise definition of the function \(f(x)\) and applying it correctly to evaluate \(f(f(-5/2))\) is crucial. The function's behavior changes based on whether \(x\) is less than or equal to \(0\) or greater than \(0\), and identifying the fractional part and absolute value expressions for \(x > 0\) is key to solving the problem.</div> </div>