PARAGRAPH 2 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 $x \in(1,3)$

JEE Advanced · 2015 · 24 May 2015 · Paper 2 · Application of Derivatives

PARAGRAPH 2
Let

F : R \to R

be a thrice differentiable function. Suppose that

F (1) = 0, F (3) = - 4

and

F^{'} (x) < 0

for all

x \in (1/2, 3)

. Let

f (x) = x F (x)

for all

x \in R

.

The correct statement(s) is(are)

Answer: (A), (B), (C)

We have

f (x) = x F (x)

, so

f^{'} (x) = F (x) + x F^{'} (x)

. At

x = 1

f^{'} (1) = F (1) + 1 \cdot F^{'} (1) = 0 + F^{'} (1) < 0

since

F^{'} (1) < 0

(A is correct). Since

F (1) = 0

and

F^{'} (x) < 0

for

x \in (1/2, 3)

F

is strictly decreasing on this interval, so

F (2) < F (1) = 0

. Thus

f (2) = 2 F (2) < 0

(B is correct). For

x \in (1, 3)

F (x) < 0

(since

F

is decreasing and

F (1) = 0

) and

F^{'} (x) < 0

. So

f^{'} (x) = F (x) + x F^{'} (x) < 0 + 0 = 0

F (x) + x F^{'} (x) < 0

. Since both terms are negative for

x > 0

f^{'} (x) < 0

throughout

(1, 3)

, so

f^{'} (x) \neq = 0

(C is correct, D is incorrect).

JEE Advanced · 2015 · 24 May 2015 · Paper 2 · One or more correct options