Paragraph: Let F: ℝ arrow ℝ be a thrice differentiable function. Suppose that F(1)=0, F(3)=-4 and F^ (x) Question: The correct statement(s) is(are)

Correct answer: A. f ^ ′ (1) < 0 F(1) = 0, F(3) = - 4 F ^ ′ (x) F ( x ) is decreasing, also F(x) Now, f(x) = xF ( x ) x ∈ ((1)/(2), 3)xF(x) ∴ f(2) Also, f ^ ′ (x) = x F ^ ′ (x) + F ( x ) f ^ ′ (1) = F ^ ′ (1) Also, f ( x ) is strictly decreasing Now, F(1) = F(x) = 0 ∴ f ^ ′ (x) ≠ 0as F ^ ′ (x) < 0andf(x) is decreasing.

JEE Advanced 2015Paper 2

Exam: JEE Advanced
Year: 2015
Paper: Paper 2
Subject: Mathematics
Chapter: Application of Derivatives

JEE Advanced 2015 · Paper 2 · Mathematics · Application of Derivatives

Paragraph: Let F: ℝ arrow ℝ be a thrice differentiable function....

Paragraph: 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}

.
Question: The correct statement(s) is(are)

Options

A ${f}^{′}\left(1\right) < 0$
B $f\left(2\right) < 0$
C ${f}^{′}\left(x\right) \ne 0foranyx \in ( 1, 3 )$
D ${f}^{′}\left(x\right) = 0forsomex \in ( 1, 3 )$

Correct answer

A. ${f}^{′}\left(1\right) < 0$

Step-by-step solution

F\left(1\right) = 0, F\left(3\right) = - 4

{F}^{&prime;}\left(x\right) < 0

F ( x )

is decreasing, also

F\left(x\right) < 0

Now,

f\left(x\right) = xF ( x )

x \in \left(\dfrac{1}{2}, 3\right)xF\left(x\right) < 0

∴ f\left(2\right) < 0

Also,

{f}^{&prime;}\left(x\right) = x{F}^{&prime;}\left(x\right) + F ( x )

{f}^{&prime;}\left(1\right) = {F}^{&prime;}\left(1\right) < 0

Also,

f ( x )

is strictly decreasing
Now,

F\left(1\right) = F\left(x\right) = 0

∴ {f}^{&prime;}\left(x\right) \ne 0as{F}^{&prime;}\left(x\right) < 0andf\left(x\right)

is decreasing.

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