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Question

# Consider the functionf(x)=(1x)2x2, where x>0At what value of x does the function attain maximum value?

A
e
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B
e
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C
1e
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D
1e
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Solution

## The correct option is D 1√eLet y=f(x)=(1x)2x2Taking natural Logaritm on both sides, we havelny=−2x2lnxNow, differentiating both sides wrt x, we have 1ydydx=−4xlnx−2x⟹dydx=(−4xlnx−2x)(1x)2x2=0⟹(−4xlnx−2x)=0⟹x=e−0.5Again differentiating wrt x, we haved2ydx2=(−4xlnx−2x)2(1x)2x2+(−4lnx−6)at e−0.5, d2ydx2=(2e−0.5−2e−0.5)2(e0.5)2e−1+(2−6)=−4<0Hence, there is a maxima at x=e−0.5.This is the required answer.

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