5 Find the absolute maximum value and the absolute minimum value of the following
functions in the given intervals:

(ii) f (x) = \sin x + \cos x , x \epsilon [0, \pi]

Answers (1)

Given function is 
f(x) = \sin x + \cos x
f^{'}(x) = \cos x - \sin x\\ f^{'}(x)= 0\\ \cos x- \sin x= 0\\ \cos = \sin x\\ x = \frac{\pi}{4}         as    x \ \epsilon \ [0,\pi]
Hence, x = \frac{\pi}{4} is the critical point  of the function f(x) = \sin x + \cos x
Now, we need to check the value of function f(x) = \sin x + \cos x  at x = \frac{\pi}{4}  and at the end points of given range  i.e. x = 0 \ and \ x = \pi
f(\frac{\pi}{4}) = \sin \frac{\pi}{4} + \cos \frac{\pi}{4}\\
            =\frac{1}{\sqrt2}+\frac{1}{\sqrt2} = \frac{2}{\sqrt2} = \sqrt2
f(0) = \sin 0 + \cos 0 = 0 + 1 = 1
f(\pi) = \sin \pi + \cos \pi = 0 +(-1) = -1
Hence, the absolute maximum value of function f(x) = \sin x + \cos x occures at x = \frac{\pi}{4} and value is \sqrt2
and absolute minimum value of function f(x) = \sin x + \cos x occurs at x = \pi and value is -1

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