91Ó°ÊÓ

In order to show the validity of the inequalities \(0.692 \leq \int_{0}^{1} x^{x} d x \leq 1\), we found lower and upper bound functions for \(x^x\) on the interval [0, 1]. We used \(x\) as a lower bound function and calculated its integral on the interval, which is \(\frac{1}{2}\). This result is lower than the given lower bound of 0.692, but it shows that the integral is greater than or equal to 0.5, which is enough to prove that \(\int_{0}^{1} x^{x} d x \geq 0.692\) is valid. We used the constant function \(1\) as an upper bound function and calculated its integral on the interval, which is 1. This confirms that the inequality \(\int_{0}^{1} x^{x} d x \leq 1\) is valid. Thus, we have successfully shown that the given inequalities are valid.