Find the Horizontal Tangent Line y=x+sin(x)

Problem

y=x+sin(x)

Solution

1. Identify the condition for a horizontal tangent line, which occurs when the derivative of the function is equal to zero.

d(y)/d(x)=0

2. Differentiate the function with respect to x using the sum rule and the derivative of the sine function.

d(y)/d(x)=1+cos(x)

3. Set the derivative equal to zero to find the xvalues where the tangent is horizontal.

1+cos(x)=0

4. Solve for x by isolating the cosine term.

cos(x)=−1

5. Determine the general solution for x where the cosine function equals −1

x=π+2*n*π

6. Substitute the xvalues back into the original function to find the corresponding yvalues.

y=(π+2*n*π)+sin(π+2*n*π)

7. Simplify the expression using the fact that sin(π+2*n*π)=0 for any integer n

y=π+2*n*π

Final Answer

y=(2*n+1)*π

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