What the maximum and minimum and what the interval is increasing and decreasing of x lnx? help me

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Answer 1 · 2017-12-27T18:20:03

#f_min=f(1/e)=1/e*ln(1/e)=-1/e#.

# f uarr" for "x gt 1/e#.

Let, #f(x)=xlnx, x in RR^+#.

#"For "f_min, f'(x)=0, and f''(x) gt 0.#

#"Now, "f(x)=xlnx rArr f'(x)=x*1/x+1*lnx=1+lnx#.

#"Also "f'(x)=1+lnx rArr f''(x)=1/x#.

#"Hence, "f'(x)=0 rArr 1+lnx=0 rArr lnx=-1 rArr x=1/e#.

#"Further, "f''(1/e)=e gt 0#.

Accordingly, #f_min=f(1/e)=1/e*ln(1/e)=-1/e#.

It also follows from the above discussion, that # f uarr" for "x gt 1/e#.