jee-advanced 2015 Q59

jee-advanced · India · paper2 Stationary points and optimisation Determine intervals of increase/decrease or monotonicity conditions

Let $F : \mathbb { R } \rightarrow \mathbb { R }$ be a thrice differentiable function. Suppose that $F ( 1 ) = 0 , F ( 3 ) = - 4$ and $F ^ { \prime } ( x ) < 0$ for all $x \in ( 1 / 2,3 )$. Let $f ( x ) = x F ( x )$ for all $x \in \mathbb { R }$. The correct statement(s) is(are)
(A) $f ^ { \prime } ( 1 ) < 0$
(B) $f ( 2 ) < 0$
(C) $f ^ { \prime } ( x ) \neq 0$ for any $x \in ( 1,3 )$
(D) $f ^ { \prime } ( x ) = 0$ for some $x \in ( 1,3 )$

Let $F : \mathbb { R } \rightarrow \mathbb { R }$ be a thrice differentiable function. Suppose that $F ( 1 ) = 0 , F ( 3 ) = - 4$ and $F ^ { \prime } ( x ) < 0$ for all $x \in ( 1 / 2,3 )$. Let $f ( x ) = x F ( x )$ for all $x \in \mathbb { R }$. The correct statement(s) is(are)\\
(A) $f ^ { \prime } ( 1 ) < 0$\\
(B) $f ( 2 ) < 0$\\
(C) $f ^ { \prime } ( x ) \neq 0$ for any $x \in ( 1,3 )$\\
(D) $f ^ { \prime } ( x ) = 0$ for some $x \in ( 1,3 )$

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