0,\\ f'(x)=(x\ln x)'=x'\ln x+x(\ln x)'=\ln x+x\cdot\frac{1}{x}=\ln x+1, \\ f'(x)=0, \ \ln x+1=0, \ln x=-1, x=e^{-1}=\frac{1}{e}, \\ x<\frac{1}{e}, \ln x<-1, \ln x+1<0, f'(x)<0, f(x)\searrow \ , \\ x>\frac{1}{e}, \ln x>-1, \ln x+1>0, f'(x)>0, f(x)\nearrow \ , \\ x_{min}=\frac{1}{e}, y_{min}=f(\frac{1}{e})=\frac{1}{e}\cdot\ln\frac{1}{e}=-\frac{1}{e}, \\ (\frac{1}{e};-\frac{1}{e})" alt="f(x)=x\ln x, \ x>0,\\ f'(x)=(x\ln x)'=x'\ln x+x(\ln x)'=\ln x+x\cdot\frac{1}{x}=\ln x+1, \\ f'(x)=0, \ \ln x+1=0, \ln x=-1, x=e^{-1}=\frac{1}{e}, \\ x<\frac{1}{e}, \ln x<-1, \ln x+1<0, f'(x)<0, f(x)\searrow \ , \\ x>\frac{1}{e}, \ln x>-1, \ln x+1>0, f'(x)>0, f(x)\nearrow \ , \\ x_{min}=\frac{1}{e}, y_{min}=f(\frac{1}{e})=\frac{1}{e}\cdot\ln\frac{1}{e}=-\frac{1}{e}, \\ (\frac{1}{e};-\frac{1}{e})" align="absmiddle" class="latex-formula">