find the coordinates of all points on the graph of f(x) = x + 2cos(x) in the interval [0, pie] at which the tangent line is horizontal.
To find the coordinates of all points on the graph of f(x) = x + 2cos(x) in the interval [0, π] at which the tangent line is horizontal, we need to find where the derivative of the function equals zero.
Step 1: Find the derivative of f(x):
f'(x) = 1 - 2sin(x)
Step 2: Set the derivative equal to zero:
1 - 2sin(x) = 0
Step 3: Solve for x:
2sin(x) = 1
sin(x) = 1/2
Step 4: Find the values of x in the interval [0, π] that satisfy sin(x) = 1/2.
The values of x in the interval [0, π] that satisfy sin(x) = 1/2 are π/6 and 5π/6.
Step 5: Find the corresponding y-coordinates by substituting the values of x into the original function f(x):
For x = π/6: f(π/6) = π/6 + 2cos(π/6) = π/6 + 2(√3/2) = π/6 + √3 = (π + 6√3)/6
For x = 5π/6: f(5π/6) = 5π/6 + 2cos(5π/6) = 5π/6 + 2(-√3/2) = 5π/6 - √3 = (5π - 6√3)/6
Therefore, the coordinates of the points on the graph of f(x) = x + 2cos(x) in the interval [0, π] at which the tangent line is horizontal are:
(π/6, (π + 6√3)/6) and (5π/6, (5π - 6√3)/6).