The Derivatives of Trigonometric Functions - Exercise 2

Exercise 2. Find the x-coordinates of all points on the graph of $f(x) = x +2\cos(x)$ in the interval $[0,\pi]$ at which the tangent line is horizontal.

Answer. The points (x,f(x)) at which the tangent line is horizontal are the ones for which f'(x) = 0. Let us first find f'(x). We have

\begin{displaymath}f'(x) = 1 - 2\sin(x)\;.\end{displaymath}

So we have to solve $1 - 2\sin(x) = 0$ which gives $\sin(x) = 1/2$. We get

\begin{displaymath}x = \frac{\pi}{6}\;\;\mbox{or}\;\; x = \pi - \frac{\pi}{6} = \frac{5\pi}{6} \cdot\end{displaymath}

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