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Question

Let the function f be defined by f(x)=xlnx, for all x>0. Then

A
f is increasing on (0,e1)
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B
f is decreasing on (0,1)
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C
The graph of f is concave down for all x
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D
The graph of f is concave up for all x
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Solution

The correct option is D The graph of f is concave up for all x
dydx=1+lnx
dydx=0x=e1=1e
It is increasing on (1e,) and decreasing on (0,1e)


Clearly, x=1e gives local minima
and f(1e)=1e
Also, d2ydx2=1x>0 (x>0)
Concave up for all x>0

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