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Question

Which of the following is (are) correct about the function f(x)=xlog x

A
f(x) is increasing on (e, )
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B
f(x) is decreasing on (0, )
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C
f(x) is increasing on (0, 1)(1, e)
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D
f(x) is decreasing on (0, 1)(1, e)
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Solution

The correct option is D f(x) is decreasing on (0, 1)(1, e)
Given the function f(x) = xlog xf'(x)=log x1(log x)2For f(x) to be an increasing function, we must havef'(x)>0log x1(log x)2>0log x1>0 [(log x)2>0 for x>0]log x>1x>ex(e, )For f(x) to be a decreasing function, we must havef'(x)<0log x1(log x)2<0log x1<0 [(log x)2>0 for x>0]log x<1x<ex(0, e) {1} [f(x) is defined for x>0 and x1]So, f(x) is decreasing on (0, 1)(1, e).

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