Find the local maxima and minima of $ h(x) = \sin x + \cos x $ in $ \left( 0, \frac{\pi}{2} \right) $

Q: Find the local maxima and minima of $ h(x) = \sin x + \cos x $ in $ \left( 0, \frac{\pi}{2} \right) $

Explanation: \[ h(x) = \sin x + \cos x \Rightarrow h'(x) = \cos x - \sin x \] \[ h'(x) = 0 \Rightarrow \cos x = \sin x \Rightarrow \tan x = 1 \Rightarrow x = \frac{\pi}{4} \] \[ h''(x) = -\sin x - \cos x < 0 \Rightarrow \text{Local maximum at } x = \frac{\pi}{4} \] \[ h\left( \frac{\pi}{4} \right) = \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} = \sqrt{2} \]

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Explanation: $$
h(x) = \sin x + \cos x \Rightarrow h'(x) = \cos x - \sin x
$$
$$
h'(x) = 0 \Rightarrow \cos x = \sin x \Rightarrow \tan x = 1 \Rightarrow x = \frac{\pi}{4}
$$
$$
h''(x) = -\sin x - \cos x < 0 \Rightarrow \text{Local maximum at } x = \frac{\pi}{4}
$$
$$
h\left( \frac{\pi}{4} \right) = \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} = \sqrt{2}
$$

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