For what values of x in 0 2 pi does the graph of fx x 2 si

For what values of x in [0, 2 pi] does the graph of f(x) = x + 2 sin x have a horizontal tangent? List the values of x below. Separate multiple values with commas. x =

Solution

f(x) = x+ 2 sin x

the function will have a horizontal tangent if slope of horizontal line is 0

so finding the slope of the function by taking the derivative

dy / dx of x + 2 sin x = 1 + 2 cos x

now setting the derivative equal to 0 and solving for x

1 + 2 cos x = 0

2 cos x = -1

cos x = -1/2

x= cos^-1 ( -1/2 )

x = 2pi/3 + 2pin , 4pi/3 + 2pin

in the interval [ 0, 2pi ]

x = 2pi/3 , 4pi/3