Example 7.1.1 If $x$ is a positive number, find the minimum value of the expression $x^{x}$.



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Consider the function $f(x)=x^{x}=e^{x \ln x}$. Its derivative is $f^{\prime}(x)=e^{x \ln x}(\ln x+1)$. Clearly, $f^{\prime}(x)=0 \Leftrightarrow \ln x=-1 \Leftrightarrow x=\frac{1}{e}$. If $x \in\left(0, \frac{1}{e}\right]$, then $f(x)$ is decreasing; if $x \in\left[\frac{1}{e},+\infty\right)$, then $f(x)$ is increasing. Therefore,
$$\min _{x \in R} f(x)=f\left(\frac{1}{e}\right)=\frac{1}{e^{\frac{1}{e}}}$$