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Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be: h(x) = sin x + cos x

Find the local maxima and local minima, if any, of the following functions. Find
also the local maximum and the local minimum values, as the case may be:

(iv) f(x) = sin x - cos x

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Given function is
h(x) = \sin x - \cos x\\ h^{'}(x)= \cos x + \sin x\\ h^{'}(x)= 0\\ \cos x + \sin x = 0\\ \cos x = -\sin x\\ x = \frac{3\pi}{4} \ \ \ \ \ \ as \ x \ \epsilon \ \left ( 0,2\pi \right )
Now, we use second derivative test
h^{''}(x)= -\sin x + \cos x\\ h^{''}(\frac{3\pi}{4}) = -\sin \frac{3\pi}{4} + \cos \frac{3\pi}{4}\\ h^{''}(3\frac{\pi}{4}) = -(\frac{1}{\sqrt2})-\frac{1}{\sqrt2}\\ h^{''}(\frac{\pi}{4})=- \frac{2}{\sqrt2} = -\sqrt2 < 0
Hence, \frac{\pi}{4}   is the point of maxima and maximum value is h\left ( \frac{3\pi}{4} \right ) which is \sqrt2
 

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