KVPY 2020-21 SB SX Exam Previous Papers

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Question : 83 of 120

Marks: +1, -0

Let

f(x)=\sin x+(x^{3}-3 x^{2}+4 x-2) \cos x

for

{x} \∈ (0,1) .

Consider the following statements
I.

f

has a zero in

(0,1)

II.

f

is monotone in

(0,1)

Then

I and II are true
I is true and II are false
I is false and II are true
I and II are false

Solution:

f(x)=\sin x+(x^{3}-3 x^{2}+4 x-2) \cos x

{x} \∈ (0,1)

f(0)=-2>0

f(1)=\sin 1 < 0

\because f(0) . f(1) < 0 \⇒f({x})

has a zero in

(0,1)

Now,

f(x)=\sin x+[(x-1)^{3}+(x-1)] \cos x

\⇒ f'(x)=(3(x-1)^{2}+2) \cos x

-\sin x[(x-1)^{3}+(x.

-1)]

=[3(x-1)^{2}+2] \cos x+[(1-x)^{3}

+(1-x)] \sin x

>0 \∀x \∈ (0,1)

\⇒f({x})

is monotone in

(0,1)

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