Limits, Continuity & Differentiability
Continuity and differentiability of functions
Grade 12
Question:
<p>If \(a^2 + b^2 + c^2 + ab + bc + ca \leq 0\), where \(a, b, c \in R\) and \(f(x) = a[x] + b|x| + c\,\text{sgn}(x)\), then in \((-2, 2)\), which of the following is <strong>not true</strong>?</p><p>[<strong>Note:</strong> \([y]\) denotes greatest integer function less than or equal to \(y\).]</p>
<p>\(f(x)\) is discontinuous at exactly two points</p>
<p>\(f(x)\) is discontinuous at exactly three points</p>
<p>\(f(x)\) is continuous and derivable for every \(x\)</p>
<p>\(f(x)\) is non-derivable at exactly one point</p>
Step-by-Step Solution
Key Concept: The constraint a² + b² + c² + ab + bc + ca ≤ 0 can be rewritten as ½[(a+b)² + (b+c)² + (c+a)²] ≤ 0, which forces a = b = c = 0 since sum of squares is non-negative. This makes f(x) = 0 (a constant function), which is continuous and differentiable everywhere.
<p><strong>Step 1:</strong> Analyze the constraint a² + b² + c² + ab + bc + ca ≤ 0.</p><p>Multiply by 2: 2a² + 2b² + 2c² + 2ab + 2bc + 2ca ≤ 0</p><p>Rewrite as: (a² + 2ab + b²) + (b² + 2bc + c²) + (c² + 2ca + a²) ≤ 0</p><p>This gives: (a+b)² + (b+c)² + (c+a)² ≤ 0</p><p><strong>Step 2:</strong> Since sum of squares is always ≥ 0, and the sum ≤ 0, we must have each square = 0.</p><p>Therefore: a + b = 0, b + c = 0, c + a = 0</p><p>Solving: a = b = c = 0</p><p><strong>Step 3:</strong> With a = b = c = 0, the function becomes f(x) = 0 for all x ∈ (-2, 2).</p><p><strong>Step 4:</strong> The zero function is:</p><ul><li>Continuous everywhere in (-2, 2) ✓</li><li>Differentiable everywhere in (-2, 2) with f'(x) = 0 ✓</li><li>Even function ✓</li><li>Bounded ✓</li></ul><p>Any statement claiming f is discontinuous, non-differentiable, or has properties other than being identically zero would be <strong>NOT TRUE</strong>.</p><p>∴ Answer: A (whichever option contradicts f(x) ≡ 0)</p>
Correct Answer: A