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 (x, y) =

Answer 1

dy/dx is the slope of the tangent line

Horizontal tangent line - slope = 0

so we need dy/dx = 0 (basically we are looking for critical points).

dy/dx = 1 + cos(x) = 0 when cos(x) = -1, ie when x = π, 3π, 5π ....

The only possible value here in the required interval is π

x=π, y = π + sin(π) = π, so point is (π,π)

You could check this by graphing the function on Desmos