<p>A point \(x = x_1\) in the domain of \(f\) is said to be a stationary point if \(f'(x_1) = 0\).</p>
<p>A point \(x = x_2\) in the domain of \(f\) is said to be a stationary point if \(f''(x_2) = 0\).</p>
<p>If \(f\) is differentiable on the open interval \((a, b)\) and if \(f\) has an absolute extremum on that interval, then it must occur at a stationary point of \(f\).</p>
<p>If \(f\) is continuous on the open interval \((a, b)\) and if \(f\) has an absolute extremum on that interval, then it must occur at a stationary point of \(f\).</p>
Step-by-Step Solution
Key Concept: A stationary point is defined by the condition that the first derivative equals zero, f'(x) = 0. This is the standard definition in calculus. Understanding the distinction between stationary points, critical points, and points where extrema occur is essential.
<p><strong>Step 1: Analyze Option A</strong></p><p>A stationary point is defined as a point x = x₁ in the domain of f where f'(x₁) = 0. This is the standard definition from calculus. At a stationary point, the tangent line is horizontal. <strong>This statement is correct by definition.</strong></p><p><strong>Step 2: Analyze Option B</strong></p><p>A point where f''(x₂) = 0 is called an inflection point (or possible inflection point), not a stationary point. The second derivative condition relates to concavity changes, not to stationary points. <strong>This is false.</strong></p><p><strong>Step 3: Analyze Option C</strong></p><p>Even if f is differentiable on the open interval (a,b), an absolute extremum may not exist on an open interval (e.g., f(x) = x on (0,1) has no maximum). If an extremum exists on an open interval and f is differentiable there, it must occur at a stationary point by Fermat's theorem. However, the premise is weak since extrema may not exist. <strong>This is conditionally true but the phrasing is problematic.</strong></p><p><strong>Step 4: Analyze Option D</strong></p><p>Continuity alone does not guarantee that if an extremum exists on an open interval (a,b), it must occur at a stationary point. For example, f(x) = |x| is continuous on (-1,1) with a minimum at x = 0, but f'(0) does not exist. The function is not differentiable at the extremum point. <strong>This is false.</strong></p><p><strong>Conclusion:</strong> Option A states the precise, universally accepted definition of a stationary point in calculus. ∴ Answer: A</p>
Correct Answer: A