Q. Let $f:(0,2 \pi) \rightarrow R$ be defined as $f(x)=\sin x \cdot e^{\sin 2 x}+\int\limits_0^x e^{\sin 2 t}(\sin t-\cos t) d t$. Then the number of points of local minima of $f(x)$ is equal to
Application of Derivatives
Solution:
We have, $ f(x)=\sin x \cdot e^{\sin 2 x} I^x \int\limits_0^x e^{\sin 2 t}(\sin t-\cos t) d t $
$\therefore f ^{\prime}( x )=\sin x (2 \cos 2 x ) \cdot e ^{\sin 2 x }+(\cos x ) e ^{\sin 2 x }+ e ^{\sin 2 x }(\sin x -\cos x ) $
$= e ^{\sin 2 x } \cdot \sin x (2 \cos 2 x +1) $
$=2 \cdot e^{\sin 2 x} \cdot \sin x \cdot\left(\cos 2 x+\frac{1}{2}\right) $
$=2 \cdot e^{\sin 2 x} \sin x \cdot 2\left(\cos ^2 x-\frac{1}{4}\right) $
$=4 \cdot e^{\sin 2 x} \sin x \cdot\left(\cos x+\frac{1}{2}\right) \cdot\left(\cos x-\frac{1}{2}\right) $
