Show that $\displaystyle f(\mathrm{x})=\tan ^{-1}(\sin \mathrm{x}+\cos \mathrm{x})$ is an increasing function in $\displaystyle \left[0, \frac{\pi}{4}\right]$.

\[\mathbf{f}^{\prime}(\mathbf{x})=\frac{\mathbf{1}}{\mathbf{1}+(\sin \mathbf{x}+\cos \mathbf{x})^{2}}(\cos \mathbf{x}-\sin \mathbf{x}) \] For $\displaystyle \mathrm{x} \in\left[\mathbf{0}, \frac{\pi}{4}\right], \cos \mathrm{x} \geq \sin \mathrm{x}$ $\displaystyle \Rightarrow \mathbf{f}^{\prime}(\mathbf{x}) \geq \mathbf{0}, \mathbf{f}$ is an increasing function in $\displaystyle \left[\mathbf{0}, \frac{\pi}{4}\right]$