Show that f(x)= (2 x+ 4 ) is an increasing function on (3 8 , 7 8… - Vidyadip

Question

Show that $f(x)=\cos \left(2 x+\frac{\pi}{4}\right)$ is an increasing function on $\left(\frac{3 \pi}{8}, \frac{7 \pi}{8}\right)$

✓

Answer

Given: $f(x)=\cos \left(2 x+\frac{\pi}{4}\right)$
$f^{\prime}(x)=-2 \sin \left(2 x+\frac{\pi}{4}\right)$
Now,
$x \in\left(\frac{3 \pi}{8}, \frac{7 \pi}{8}\right)$
$\Rightarrow \quad \frac{3 \pi}{4}<2 x<\frac{7 \pi}{4}$
$\Rightarrow \quad \frac{\pi}{4}+\frac{3 \pi}{4}<2 x+\frac{\pi}{4}<\frac{7 \pi}{4}+\frac{\pi}{4}$
$\Rightarrow \quad \pi<2 x+\frac{\pi}{4}<2 \pi$
$\Rightarrow \quad \sin \left(2 x+\frac{\pi}{4}\right)<0$
$\Rightarrow \quad-2 \sin \left(2 x+\frac{\pi}{4}\right)>0$
$\Rightarrow \quad f^{\prime}(x)>0$
Hence, $f(x)$ is increasing on $\left(\frac{3 \pi}{8}, \frac{7 \pi}{8}\right)$

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