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Mathematics · 2025 · 3 marks
CBSE 2025 · Region 5 · Set 2 · Q26
Show that f( x )= tan^-1( sin x + cos x ) is an increasing function in [0, (π)/4].
Show that $\displaystyle f(\mathrm{x})=\tan ^{-1}(\sin \mathrm{x}+\cos \mathrm{x})$ is an increasing function in $\displaystyle \left[0, \frac{\pi}{4}\right]$.
Marking-scheme solution
\[\mathbf{f}^{\prime}(\mathbf{x})=\frac{\mathbf{1}}{\mathbf{1}+(\sin \mathbf{x}+\cos \mathbf{x})^{2}}(\cos \mathbf{x}-\sin \mathbf{x})
\]
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 $\displaystyle \left[\mathbf{0}, \frac{\pi}{4}\right]$
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