Applications of Derivatives
Stationary Points and Extrema
Grade 12
Question:
<p>A point \( x = x_1 \) in the domain of \( f \) is said to be a stationary point if \( f'(x_1) = 0 \). Which of the following statements is/are true?<br>(1) Every local maximum or minimum of \( f \) is a stationary point.<br>(2) Every stationary point of \( f \) is a local maximum or minimum.<br>(3) If \( f \) has an absolute maximum on an open interval \((a,b)\), it must occur at a stationary point.<br>(4) \( f \) is continuous on \((a,b)\) implies \( f \) has an absolute extremum on \((a,b)\).</p>
<p>Only (1) is true</p>
<p>Only (2) is true</p>
<p>Only (3) is true</p>
<p>Only (4) is true</p>
Step-by-Step Solution
Key Concept: A stationary point is necessary for local extrema at interior points (by Fermat's theorem), but not sufficient—and extrema on open intervals may not exist if the function isn't bounded or is unbounded near endpoints.
<p><strong>Statement (1) - TRUE:</strong> If f has a local maximum or minimum at an interior point x₁ ∈ (a,b) where f is differentiable, then by Fermat's Theorem, f'(x₁) = 0. Thus x₁ is a stationary point.</p><p><strong>Statement (2) - FALSE:</strong> Counterexample: f(x) = x³ at x = 0. Here f'(0) = 0 (stationary point), but x = 0 is neither a local maximum nor minimum—it's an inflection point.</p><p><strong>Statement (3) - TRUE:</strong> If f has an absolute maximum at some point x₁ ∈ (a,b) on an open interval where f is differentiable, then x₁ is also a local maximum. By Fermat's Theorem, f'(x₁) = 0, so x₁ must be a stationary point.</p><p><strong>Statement (4) - FALSE:</strong> Counterexample: f(x) = x is continuous on (0,1) but has no absolute maximum or minimum on this open interval (supremum is 1, infimum is 0, but neither is attained).</p><p>∴ Answer: Statements (1) and (3) are true → <strong>C</strong></p>
Correct Answer: C