Let f : R ? R be a thrice differentiable function such that f ( 0 ) = 0 , ? f ( 1 ) = 1 , ? f ( 2 ) = - 1 , ? f ( 3 ) = 2 and f ( 4 ) = - 2 . Then, the minimum number of zeros of ( 3 f ' f ' ' + f f ' ' ' ) ( x ) is _______ . [2024]

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Solution

Q.
Let $f : R \to R$ be a thrice differentiable function such that $f (0) = 0, f (1) = 1, f (2) = - 1, f (3) = 2 and f (4) = - 2 .$ Then, the minimum number of zeros of $(3 f' f'' + f f''') (x)$ is _______ . [2024]

Ans.
(5)

$∵ f : R \to R and f (0) = 0, f (1) = 1, f (2) = - 1, f (3) = 2 and f (4) = - 2, then$

$∴ f (x) has at least 4 real roots.$

$Then f' (x) has atleast 3 real roots and f'' (x) has atleast 2 real roots.$

$\frac{d}{d x} (f^{3} \cdot f'') = 3 f^{2} \cdot f' \cdot f'' + f^{3} \cdot f''' = f^{2} (3 f' \cdot f'' + f \cdot f''') .$

$Hence, f^{3} . f'' has at least 6 roots.$

$Then its differentiation has at least 5 distinct roots.$

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