Horizontal Tangent Line

Question

Determine the point at which the graph of the function has a horizontal tangent line. (If the function has no horizontal tangent line, enter NONE.)

y = x + sin(x), 0 ≤ x < 2𝜋

Answer 1

y=x +sin(x)

take the derivative and set=0

y'=1+cos(x)=0

cos(x)=-1

x= cos^-1(-1)

x=pi radians

y = pi+sin(pi) = pi+0=pi

the point (pi, pi) has a horizontal tangent line

If you used the older or looser definition of tangent line, you'd be falsely led to say DNE, Does not Exist, as there is no line which just touches the curve and does not intersect it. That would exclude inflection points. But the modern definition includes inflection points where the curve's derivative = 0. (pi,pi) is an inflection pont where the concavity changes from concave down to concave up

use desmos graphing, the curve looks momentarily flat at about x= 3.14

or if x is measured in degrees, then

x=180

y=180

the point (180,180)