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Question

The function fx=log1+x-2x2+x is increasing for all values of x, then


A

0,

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B

-,0

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C

-,

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D

None of these

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Solution

The correct option is A

0,


Explanation for correct option

Step 1. Determine the interval of the function

The given function is, fx=log1+x-2x2+x

Differentiate with respect to x.

f'x=ddxlog1+x-22+x

Use sum/difference rule f±g'=f'±g', where f and g are the function.

f'x=ddxln1+x-ddx2x2+x

Use chain rule

f'x=11+xddx1+x-12x2+xddx2x2+x

Derivative of constant ddxa=0 and common derivative ddxx=1.

f'x=11+x×1-212+x-1×x(2+x)2

=11+x-42+x2

Step 2 simplify the equation

We will simplify the equation 11+x-42+x2.

f'x=x+22-4x+1x+1x+22

=x2+4x+4x+1x+22

=x2+4x-4x+4-4x+1x+22

=x2x+1x+22

Therefore, f'x>0

By substituting the values,

x2x+1x+22>0, where 1+x>0

Thus, fx is increasing function on 0, for all x>-1.

Hence, option A is correct answer.


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