Question

For what values of\(x\)does the graph of\(f\)have a horizontal tangent?

\(f\left( x \right) = x + 2{\rm{sin}}x\)

Short answer

The required value is \(x = \left( {2n + 1} \right)\pi \pm \frac{\pi }{3},\,\,\,\,\,\,\,{\rm{for}}\,\,n \in {\rm I}\).

Step-by-step solution

Horizontal Tangents of Trigonometric Functions

When any trigonometric function is subjected to have horizontal tangents, then the derivative of that trigonometric function will be equal to zero.

Differentiation of the given function and solving for \(x\)

Thegiven function is\(f\left( x \right) = x + 2\sin x\).

Differentiating the given function with respect to\(x\)and equating it to zero, we get:

\(\begin{aligned}\frac{d}{{dx}}\left( {f\left( x \right)} \right) &= 0\\\frac{d}{{dx}}\left( {x + 2\sin x} \right) &= 0\\1 + 2\cos x &= 0\\\cos x &= - \frac{1}{2}\\x &= \left( {\frac{{2\pi }}{3} + 2\pi n} \right)\,\,{\rm{or }}\left( {\frac{{4\pi }}{3} + 2\pi n} \right)\,\,\,\,\,\,\forall n \in {\rm I}\\x &= \left( {2n + 1} \right)\pi \pm \frac{\pi }{3}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\forall n \in {\rm I}\end{aligned}\)

Hence, the required answer is \(x = \left( {2n + 1} \right)\pi \pm \frac{\pi }{3},\,\,\,\,\,\,\,{\rm{for}}\,\,n \in {\rm I}\).