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The maximum value of the real valued function $f(x)=\cos x + \sin x$ is

  1. $2$
  2. $1$
  3. $0$
  4. $\sqrt{2}$
in Calculus by Veteran (431k points)
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1 Answer

+1 vote

Answer: $\mathbf D$


Method 1:

Maximum and minimum values of the expression of the form

$a\sin x + b \cos x$ is given by

$\pm \sqrt{a^2+b^2}$

Here, $a = 1, \;and \; b = 1$

So, Maximum value $= \sqrt{1^2 + 1^2} = \sqrt2$

So, option $\mathbf D$ is the correct answer.

Method 2:

Let $y = \cos x + \sin x$

On differentiating with respect to x, we get:

$\frac{dy}{dx} = -\sin x + \cos x$

Put this equation equals to 0, for calculating the maxima.


$\frac{dy}{dx}  = 0 \implies \sin x = \cos x \implies x = \frac{\pi}{4}$

So, we will get the maximum value at $x = \frac{\pi}{4} = \frac{1}{\sqrt 2} = \sqrt 2$

So, again by this method option $\mathbf D$ is the right answer.

by Boss (18.9k points)
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