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Question

Which one is the correct statement about the function fx=sin2x


A

fx is increasing in 0,π2 and decreasing in π2,π

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B

fx is decreasing in 0,π2 and increasing in π2,π

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C

fx is increasing in 0,π4 and decreasing in π4,π2

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D

The statements (a), (b) and (c) are all correct

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Solution

The correct option is C

fx is increasing in 0,π4 and decreasing in π4,π2


Explanation for the correct option:

Step 1: Calculating differentiation of given function

Given: fx=sin2x

Calculating the derivative of the function,

f'x=2cos2x ddxcos2x=2sin2x

Step 2: Finding the increasing interval

A function fx is increasing in the interval a,b if f'x>0 in the interval a,b.

When 0<cos2x<1 , then f'x>0.

We know that,

0cos2x1 when 02xπ2

0cos2x1 when 02xπ2×2

0cos2x1 when 0xπ4

So, the function is increasing when x0,π4

Step 3: Finding the decreasing interval

A function fx is decreasing in the interval a,b if f'x<0 in the interval a,b.

When -1<cos2x<0 , then f'x<0.

We know that,

-1cos2x0 when π22xπ

-1cos2x0 when π2×2xπ2

-1cos2x0 when π4xπ2

So, the function is decreasing when xπ4,π2

Hence, option (C) is correct.


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