Let f(x) be a continuous function in x∈[0,5] and twice differentiable function in x∈(0,5) such that f(4)=f(5)=0 then which of the following is/are correct

There exist α ∈ ( 0 , 5 ) such that f ' ' ( α ) = 0 , There exist some α ∈ ( 0 , 4 ) such that n f ( α ) + α f ' ( α ) = 0 , n ∈ N , There exist α 1 , α 2 ∈ ( 0 , 5 ) such that 4 f ' ( α 1 ) + f ' ( α 2 ) + f ( 0 ) = 0 ( α 1 < α 2 )

Let f(x) be a continuous function in x∈[0,5] and twice differentiable function in x∈(0,5) such that f(4)=f(5)=0 then which of the following is/are correct

Let g ( x ) = x n f ( x ) ( g ( 4 ) = g ( 0 ) = g ( 5 ) = 0 ) g ' ( x ) = x n f ' ( x ) + f ( x ) n . x n − 1 g ' ( α ) = 0 ⇒ n f ( α ) + α f ' ( α ) = 0 BY LMVT on [0,4] & [4,5] f ' ( C 1 ) = f ( 4 ) − f ( 0 ) 4 , f ' ' ( C 2 ) = f ( 5 ) − f ( 4 ) 1 = − f ( 0 ) 4 , f ' ( C 2 ) = 0 4 f ' ( C 1 ) = − f ( 0 ) 4 f ' ( C 1 ) + f ( 0 ) + f ' ( C 2 )

There exist α ∈ ( 0 , 5 ) such that f ' ' ( α ) = 0

There exist α ∈ ( 0 , 5 ) such that n f ( α ) + α f ' ( α ) = 0 , n ∈ N

There exist some α ∈ ( 0 , 4 ) such that n f ( α ) + α f ' ( α ) = 0 , n ∈ N

There exist α 1 , α 2 ∈ ( 0 , 5 ) such that 4 f ' ( α 1 ) + f ' ( α 2 ) + f ( 0 ) = 0 ( α 1 < α 2 )

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Let $f (x)$ be a continuous function in $x \in [0, 5]$ and twice differentiable function in $x \in (0, 5)$ such that $f (4) = f (5) = 0$ then which of the following is/are correct

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There exist $α \in (0, 5)$ such that $f^{''} (α) = 0$

There exist $α \in (0, 5)$ such that $n f (α) + α f^{'} (α) = 0, n \in N$

There exist some $α \in (0, 4)$ such that $n f (α) + α f^{'} (α) = 0, n \in N$

There exist $α_{1}, α_{2} \in (0, 5)$ such that $4 f^{'} (α_{1}) + f^{'} (α_{2}) + f (0) = 0 (α_{1} < α_{2})$

answer is A, C, D.

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Detailed Solution

Let $g (x) = x^{n} f (x) (g (4) = g (0) = g (5) = 0)$
$g^{'} (x) = x^{n} f^{'} (x) + f (x) n . x^{n - 1}$
$g^{'} (α) = 0 \Rightarrow n f (α) + α f^{'} (α) = 0$
BY LMVT on [0,4] & [4,5]
$f^{'} (C_{1}) = \frac{f (4) - f (0)}{4}$ , $f^{''} (C_{2}) = \frac{f (5) - f (4)}{1}$
= $- \frac{f (0)}{4}$ , $f^{'} (C_{2}) = 0$
$4 f^{'} (C_{1}) = - f (0)$
$4 f^{'} (C_{1}) + f (0) + f^{'} (C_{2})$