<p>If <span class='math'>\(f(x)\)</span> is continuous and derivable, <span class='math'>\(\forall x \in \mathbb{R}\)</span> and <span class='math'>\(f'(c) = 0\)</span> for exactly 2 real values of 'c', then the number of real and distinct values of 'd' for which <span class='math'>\(f(d) = 0\)</span> can be</p>
Step-by-Step Solution
Key Concept: By Rolle's theorem, if f(x) = 0 has n real roots, then f'(x) = 0 has at least (n-1) real roots. Conversely, with 2 critical points, f can have 1, 2, or 3 zeros depending on its shape.
<p><strong>Key Principle:</strong> If <span class='math'>$f'(x) = 0$</span> has n real roots, then <span class='math'>$f(x) = 0$</span> has at most (n + 1) real roots.</p><p><strong>Given:</strong> <span class='math'>$f'(c) = 0$</span> for exactly 2 real values of c.</p><p><strong>Step 1:</strong> With 2 critical points, the function f(x) can have at most 3 real roots. This depends on the behavior of f at critical points and endpoints.</p><p><strong>Step 2:</strong> Consider different cases:</p><p> (a) If f has 2 local extrema and crosses x-axis once: <span class='math'>$f(d) = 0$</span> has 1 real root</p><p> (b) If f crosses x-axis at two points: <span class='math'>$f(d) = 0$</span> has 2 real roots</p><p> (c) If f crosses x-axis at three points: <span class='math'>$f(d) = 0$</span> has 3 real roots</p><p>∴ Answers are (a), (b), and (c).</p>
Correct Answer: A, B, C