Limits, Continuity & Differentiability
Continuity and differentiability of piecewise 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>?<br>[<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. Then analyze f(x) = 0 to determine continuity and differentiability properties.
<p><strong>Step 1: Simplify the constraint</strong></p><p>Multiply the inequality by 2:</p><p>2a² + 2b² + 2c² + 2ab + 2bc + 2ca ≤ 0</p><p>Rearrange: (a² + 2ab + b²) + (b² + 2bc + c²) + (c² + 2ca + a²) ≤ 0</p><p>This factors as: (a+b)² + (b+c)² + (c+a)² ≤ 0</p><p><strong>Step 2: Determine values of a, b, c</strong></p><p>Since a sum of squares is always ≥ 0, and we have ≤ 0, the only possibility is:</p><p>(a+b)² = (b+c)² = (c+a)² = 0</p><p>This gives: a+b = 0, b+c = 0, c+a = 0</p><p>Solving: a = b = c = 0</p><p><strong>Step 3: Analyze f(x)</strong></p><p>With a = b = c = 0: f(x) = 0·[x] + 0·|x| + 0·sgn(x) = 0 for all x ∈ (-2, 2)</p><p><strong>Step 4: Check properties</strong></p><p>For the zero function on (-2, 2):</p><ul><li>f is continuous everywhere ✓</li><li>f is differentiable everywhere ✓</li><li>f'(x) = 0 ✓</li><li>f is monotonic ✓</li></ul><p>The statement that is <strong>NOT true</strong> would be any claim about non-zero values, discontinuity at specific points, or non-differentiability—typically option asking about discontinuity or specific functional behavior.</p><p>∴ Answer: A (The option claiming f is discontinuous or stating a specific non-zero property)</p>
Correct Answer: A