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Question

The function x1-x2,(x>0) has


A

A local maxima

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B

A local minima

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C

Neither local maxima nor a local minima

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D

None of the above

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Solution

The correct option is A

A local maxima


The explanation for the correct answer.

Step 1: Solve for the point of extremum of a function x1-x2,(x>0)

f(x)=x1-x2

ddxu×v=u×dvdx+v×dudx

f'(x)=x×12×(-2x)1-x2+1-x2f'(x)=1-2x21-x2

Equating f'x with 0 we get

1-2x21-x2=0

x=±12

But as x>0,we have x=12

Step 2: Solve for the second derivative.

f''(x)=1-x2-4x-1-2x2-x1-x21-x2f''(x)=1-x2(-4x)-(1-2x2)(-x)(1-x2)1-x2f''(x)=2x3-3x1-x232f''(12)=-2122=-4f''(12)=-ve ...ddxuv=v×dudx-udvdxv2

Hence, conditions for maxima are satisfied

Then f(x) is maximum at x=12

Hence, option(A) is the correct answer.


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