9. What is the maximum value of the function \sin x + \cos x

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 = 2n\pi+\frac{\pi}{4} \ where \ n \ \epsilon \ I       
Hence, x = 2n\pi+\frac{\pi}{4} is the critical point  of the function f(x) = \sin x + \cos x
Now, we need to check the value of the function f(x) = \sin x + \cos x  at x = 2n\pi+\frac{\pi}{4} 
Value is same for all cases so let assume that n = 0
Now
f(\frac{\pi}{4}) = \sin \frac{\pi}{4} + \cos \frac{\pi}{4}\\
            =\frac{1}{\sqrt2}+\frac{1}{\sqrt2} = \frac{2}{\sqrt2} = \sqrt2


Hence, the maximum value of the function f(x) = \sin x + \cos x  is \sqrt2
 

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