Find the absolute maximum value and the absolute minimum value of… - Vidyadip

Question

Find the absolute maximum value and the absolute minimum value of the function:
f(x) = sin x + cos x , x $\in$ [0, $\pi$]

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Answer

It is given that f(x) = sin x + cos x, $x \in[0, \pi]$
f'(x) = cos x - sin x
Now, f'(x) = 0
$\Rightarrow$ cos x - sin x = 0
$\Rightarrow$ cos x = sin x
$\Rightarrow$ tan x = 1
$\Rightarrow x=\frac{\pi}{4}$
Now, we evaluate the value of 'f ' at critical point $x=\frac{\pi}{4}$ and at end points of the interval [0, $\pi$]
$f\left(\frac{\pi}{4}\right)=\sin \frac{\pi}{4}+ \cos \frac{\pi}{4}=\left(\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}\right)=\frac{2}{\sqrt{2}}=\sqrt{2}$
f(0) = sin 0 + cos 0 = 0 + 1 = 1
$f(\pi)=\sin \pi+\cos \pi=0-1=-1$
Therefore, the absolute maximum value of f on $[0, \pi]$ is $\sqrt{2}$ occuring at $x=\frac{\pi}{4}$
And, the absolute minimum value of f on $[0, \pi]$ is -1 occurring at x = $\pi$

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