\noindent\textbf{1}

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Consider the following function $f(x)$.

$$f(x) = (\cos x)\log(\cos x) - \cos x + \int_0^x (\cos t)\log(\cos t)\, dt \quad \left(0 \leq x < \frac{\pi}{2}\right)$$

\begin{itemize}
    \item[(1)] Show that $f(x)$ has a minimum value on the interval $0 \leq x < \dfrac{\pi}{2}$.
    \item[(2)] Find the minimum value of $f(x)$ on the interval $0 \leq x < \dfrac{\pi}{2}$.
\end{itemize}



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