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Separate the interval of convaity of y =...

Separate the interval of convaity of y =x `log_(e)x-(x^(2))/(2)+(1)/(2)`

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The correct Answer is:
Concave upward in (0,1)
Concave downward in `(1,oo)`
`y=1+log_(e)x-x`
`rarr y''=(1)/(x)-1=(1-x)/(x)`
`y'' gt for x in (0,1)`
and `y'' lt 0 for x in (1,oo)`
`therefore ` graph is concave upward for x in (0,1) and concave downward for x in (1,o)`
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