What is the absolute minimum of #f(x)=xlnx#?

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Answer 1 · 2016-01-12T09:11:27

Minimum Point at #(1/e, -1/e)#

the given #f(x) = x* ln x#

obtain the first derivative #f' (x)# then equate to zero.

#f' (x) = x*(1/x) + ln x * 1 = 0#

#1 + ln x = 0#

#ln x= -1#

#e^-1=x#

#x=1/e#

Solving for #f(x) # at #x= 1/e#

#f(x)=(1/e)*ln (1/e)#

#f(x)=(1/e)*(-1)#

#f(x)=-1/e#

so the point #(1/e, -1/e)# is located at the 4th quadrant which is a minimum point.