Question

How do you find the points where the graph of the function `f(x)=sin2x+sin^2x` has horizontal tangents?

Answer 1

Horizontal tangent means neither increasing nor decreasing. Specifically, the derivative of the function has to be zero `f'(x)=0`.

Explanation:

`f(x)=sin(2x)+sin^2x`

`f'(x)=cos(2x)(2x)'+2sinx*(sinx)'`

`f'(x)=2cos(2x)+2sinxcosx`

Set `f'(x)=0`

`0=2cos(2x)+2sinxcosx`

`2sinxcosx=-2cos(2x)`

`sin(2x)=-2cos(2x)`

`sin(2x)/cos(2x)=-2`

`tan(2x)=-2`

`2x=arctan(2)`

`x=(arctan(2))/2`

`x=0.5536`

This is one point. Since solution was given out by `tan` , other points will be every π times the factor in `2x` meaning `2π`. So the points will be:

`x=0.5536+2n*π`

Where `n` is any integer.
graph{sin(2x)+(sinx)^2 [-10, 10, -5, 5]}