Identifying Horizontal Tangent Lines

Find all points on the graph of \(f(x) =x+\sin x\) where the tangent line is horizontal.

Solution Since tangent lines are horizontal at points on the graph of \(f\) where \(f^\prime (x) =0,\) we begin by finding \(f^\prime\): \[f^\prime (x) =1+\cos x\] Now we solve the equation \(f^\prime (x) =0.\) \begin{array}{rcl@{\qquad}l} f^\prime (x) &=& 1+\cos x=0 \\[0pt] \cos x &=& -1 \\[0pt] x &=& (2k+1) \pi &\hbox{where }{k}\hbox{ is an integer} \end{array}

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The graph of \(f( x) =x+\sin x\) has a horizontal tangent line at each of the points \(( ( 2k+1) \pi , ( 2k+1) \pi )\), \(k\) an integer. See Figure 27 for the graph of \(f\) with the horizontal tangent lines marked. Notice that each of the points with a horizontal tangent line lies on the line \(y=x.\)