The function f(x) = tan^{-1}(sin x + cos x) i | vedclass

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(D) Let $g(x) = \sin x + \cos x = \sqrt{2} \sin(x + \pi/4)$.Since $f(x) = 	an^{-1}(g(x))$ and $	an^{-1}(u)$ is an increasing function, $f(x)$ increa

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$(-\pi/2, \pi/4)$

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(D) Let $g(x) = \sin x + \cos x = \sqrt{2} \sin(x + \pi/4)$.Since $f(x) = 	an^{-1}(g(x))$ and $	an^{-1}(u)$ is an increasing function, $f(x)$ increases when $g(x)$ increases.$g'(x) = \cos x - \sin x$.For $g(x)$ to be increasing, $g'(x) > 0 \implies \cos x > \sin x$.In the interval $(-\pi/2, \pi/2)$, $\cos x > \sin x$ holds when $x Thus, the function is increasing in $(-\pi/2, \pi/4)$.

The function $f(x) = \tan^{-1}(\sin x + \cos x)$ is an increasing function in the interval.....

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