JEE Mains · Maths · STD 12 - 5. continuity and differentiation
Let \(f\) be a real polynomial of degree \(n\) such that \(f(x) = f'(x) f''(x)\), for all \(x \in \mathbb{R}\). If \(f(0) = 0\), then \(36\left(f'(2) + f''(2) + \int_0^2 f(x)\,dx\right)\) is equal to:
- A \(42\)
- B \(46\)
- C \(56\)
- D \(66\)
Answer & Solution
Correct Answer
(C) \(56\)
Step-by-step Solution
Detailed explanation
Let the degree of the polynomial \(f(x)\) be \(n\). The degree of \(f'(x)\) is \(n-1\) and the degree of \(f''(x)\) is \(n-2\). Since \(f(x) = f'(x) f''(x)\), equating the degrees on both sides gives: \(n = (n-1) + (n-2) \Rightarrow n = 3\) Let \(f(x) = ax^3 + bx^2 + cx + d\).…
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