Question
The point in the interval $$\left( {0,\,2\pi } \right)$$ where $$f\left( x \right) = {e^x}\sin \,x$$ has maximum slope is :
A.
$$\frac{\pi }{4}$$
B.
$$\frac{\pi }{2}$$
C.
$$\pi $$
D.
$$\frac{{3\pi }}{2}$$
Answer :
$$\frac{\pi }{4}$$
Solution :
$$\eqalign{
& {\text{Given, }}\,f\left( x \right) = {e^x}\sin \,x \cr
& \Rightarrow f'\left( x \right) = {e^x}\cos \,x + {e^x}\sin \,x \cr
& \Rightarrow {\text{ slope}} = {e^x}\left( {\cos \,x + \sin \,x} \right) \cr
& {\text{Now, }}\frac{d}{{dx}}\left( {\cos \,x + \sin \,x} \right) = 0 \cr
& \Rightarrow - \sin \,x + \cos \,x = 0 \cr
& \Rightarrow \sin \,x = \cos \,x \cr
& \Rightarrow \tan \,x = 1 \cr
& \Rightarrow x = \frac{\pi }{4} \cr} $$