Given:
The function is
[tex]y=cos2x-5cosx[/tex]Required:
To find the points at which the tangent equations to the graph of the following functions are parallel to the x-axis.
Explanation:
Differentiate the given function.
[tex]\begin{gathered} \frac{dy}{dx}=2(-sin2x)-5(-sinx) \\ \frac{dy}{dx}=-2sin2x+5sinx \end{gathered}[/tex]Given that tangent to the curve is parallel to the x-axis.
So the slope of the tangent = Slope of X-axis
[tex]\begin{gathered} \frac{dy}{dx}=0 \\ 5sinx-2sin2x=0 \end{gathered}[/tex]Use the identity
[tex]sin2x=2sinxcosx[/tex]Now
[tex]\begin{gathered} 5sinx-2(2sinxcosx)=0 \\ 5sinx-4sinxcosx=0 \end{gathered}[/tex]Take out common sinx
[tex]sinx(5-4cosx)=0[/tex][tex]undefined[/tex]