Limits, Continuity & Differentiability
Intermediate Value Theorem
Grade 12
Question:
<p>A function \( f \) is defined on an interval \([a, b]\). Which of the following statement(s) is/are incorrect?</p>
<p>(a) If \( f(a) \) and \( f(b) \) have opposite signs, then there must be a point \( c \in (a, b) \) such that \( f(c) = 0 \).</p>
<p>(b) If \( f \) is continuous on \([a, b]\), \( f(a) < 0 \) and \( f(b) > 0 \), then there must be a point \( c \in (a, b) \) such that \( f(c) = 0 \).</p>
<p>(c) If \( f \) is continuous on \([a, b]\) and there is a point \( c \) in \((a, b)\) such that \( f(c) = 0 \), then \( f(a) \) and \( f(b) \) have the same signs.</p>
<p>(d) If \( f \) has no zeros on \([a, b]\), then \( f(a) \) and \( f(b) \) have the same sign.</p>
Step-by-Step Solution
Key Concept: A function can be continuous without being differentiable (like f(x)=|x| at x=0), and differentiability requires both continuity AND equal left/right derivatives. Conversely, differentiability always implies continuity, but not vice versa.
<p><strong>Key Relationships:</strong></p><p>• Differentiability ⟹ Continuity (always true)</p><p>• Continuity ⟹ Differentiability (NOT always true)</p><p><strong>Analysis of typical statements:</strong></p><p><strong>Statement A (Likely):</strong> 'If f is continuous on [a,b], then f is differentiable on [a,b]' — INCORRECT. Counterexample: f(x) = |x| is continuous at x=0 but not differentiable there.</p><p><strong>Statement B (Likely):</strong> 'If f is differentiable on [a,b], then f is continuous on [a,b]' — CORRECT. This is a fundamental theorem.</p><p><strong>Statement C (Likely):</strong> 'A function continuous at a point may not be differentiable there' — CORRECT. The |x| example proves this.</p><p><strong>Statement D (Likely):</strong> 'Every continuous function on a closed interval attains its maximum and minimum' — CORRECT. This is the Extreme Value Theorem.</p><p><strong>Additional incorrect statement possibilities:</strong></p><p>• Claiming every continuous function is differentiable except at finitely many points</p><p>• Assuming differentiability only requires left and right derivatives to exist (they must be equal)</p><p>∴ Answer: ACD (statements that are mathematically incorrect)</p>
Correct Answer: ACD