Find the Critical Points x-2sin(x)

Problem

ƒ(x)=x−2*sin(x)

Solution

1. Define the function as ƒ(x)=x−2*sin(x)

2. Find the derivative of the function with respect to x to determine the rate of change.

d(ƒ(x))/d(x)=1−2*cos(x)

3. Set the derivative to zero to find the values of x where the slope of the tangent line is horizontal, which defines the critical points.

1−2*cos(x)=0

4. Isolate the trigonometric term by subtracting 1 from both sides and then dividing by -2.

−2*cos(x)=−1

cos(x)=1/2

5. Solve for x using the unit circle or inverse trigonometric functions. The cosine function equals 1/2 at specific angles within the standard interval [0,2*π)

x=π/3

x=(5*π)/3

6. Generalize the solution to account for the periodicity of the cosine function, where n is any integer.

x=π/3+2*n*π

x=(5*π)/3+2*n*π

Final Answer

x=π/3+2*n*π,(5*π)/3+2*n*π

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