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JEE Advanced 2012Paper 2

Exam: JEE Advanced
Year: 2012
Paper: Paper 2
Subject: Mathematics
Chapter: Application of Derivatives

JEE Advanced 2012 · Paper 2 · Mathematics · Application of Derivatives

If f(x)=∫_ 0 ^ x e^ t^ 2 (t-2)(t-3) d t for all x ∈(0, ∞) , then

If

f(x)=\int_{0}^{x} e^{t^{2}}(t-2)(t-3) d t

for all

x \in(0, \infty)

, then

Options

A $f$ has a local maximum at $x=2$
B $f$ is decreasing on $(2,3)$
Cthere exists some $c \in(0, \infty)$ , such that $f^{\prime \prime}(c)=0$
D $f$ has a local minimum at $x=3$

Correct answer

A. $f$ has a local maximum at $x=2$

Step-by-step solution

f(x)=\int_{0}^{x} e^{t^{2}}(t-2)(t-3) d t

\Rightarrow f^{\prime}(x)=e^{x^{2}} \cdot(x-2)(x-3)

Put

f^{\prime}(x)=0 \Rightarrow x=2,3

f^{\prime \prime}(x)=e^{x^{2}} \cdot 2 x\left(x^{2}-5 x+6\right)+e^{x^{2}}(2 x-5)

f^{\prime \prime}(2)=-

ve and

f^{\prime \prime}(3)=+

ve

\therefore \quad x=2

is a point of local maxima. and

x=3

is a point of local minima. Also for

x \in(2,3), f^{\prime}(x) < 0

\Rightarrow \quad f

is decreasing on

(2,3)

. Also we observe

f^{\prime \prime}(0) < 0

and

f^{\prime \prime}(1)>0

\therefore \quad

There exists some

C \in(0,1)

such that

f^{\prime \prime}(C)=0

Hence all the options are correct.

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