<p><strong>Paragraph for Question nos. 638 and 639</strong><br>Let \(f(x) = x^2 - 5x + 6\), \(g(x) = f(|x|)\), \(h(x) = |g(x)|\).</p><p>The number of integral values of \(x\) satisfying the equation \(g(x) + |g(x)| = 0\) is equal to:</p>
Step-by-Step Solution
Key Concept: The equation g(x) + |g(x)| = 0 simplifies to |g(x)| = -g(x), which holds only when g(x) ≤ 0. You must find where f(|x|) is non-positive, considering that f is a parabola with roots at 2 and 3.
<p><strong>Step 1:</strong> Analyze the equation g(x) + |g(x)| = 0</p><p>This equation holds when |g(x)| = -g(x), which is true if and only if g(x) ≤ 0.</p><p><strong>Step 2:</strong> Express g(x) in terms of x</p><p>Since g(x) = f(|x|) and f(x) = x² - 5x + 6:</p><p>g(x) = |x|² - 5|x| + 6 = (|x| - 2)(|x| - 3)</p><p><strong>Step 3:</strong> Solve g(x) ≤ 0</p><p>We need (|x| - 2)(|x| - 3) ≤ 0</p><p>This holds when 2 ≤ |x| ≤ 3</p><p><strong>Step 4:</strong> Convert to condition on x</p><p>2 ≤ |x| ≤ 3 means: -3 ≤ x ≤ -2 or 2 ≤ x ≤ 3</p><p><strong>Step 5:</strong> Count integral values</p><p>From [-3, -2]: x ∈ {-3, -2} → 2 values</p><p>From [2, 3]: x ∈ {2, 3} → 2 values</p><p>Total integral values = 4</p><p>∴ Answer: C</p>
Correct Answer: C