Question

for what values of x does the graph of f have a horizontal tangent? (use n as your integer variable. enter your answers as a comma - separated list.) f(x)=x - 2 sin(x)

Explanation:

Step1: Recall derivative - tangent relation

A horizontal tangent occurs when $f'(x)=0$. First, find the derivative of $f(x)=x - 2\sin(x)$. Using the sum - difference rule and the derivatives of basic functions ($\frac{d}{dx}(x)=1$ and $\frac{d}{dx}(\sin(x))=\cos(x)$), we have $f'(x)=\frac{d}{dx}(x)-2\frac{d}{dx}(\sin(x))$.
$f'(x)=1 - 2\cos(x)$.

Step2: Set derivative equal to 0

Set $f'(x)=0$. So, $1 - 2\cos(x)=0$.
Rearrange the equation to solve for $\cos(x)$:
$2\cos(x)=1$, then $\cos(x)=\frac{1}{2}$.

Step3: Find x - values

The general solution for $\cos(x)=\frac{1}{2}$ is $x = 2n\pi\pm\frac{\pi}{3}$, where $n\in\mathbb{Z}$.

Answer:

$x = 2n\pi+\frac{\pi}{3},2n\pi - \frac{\pi}{3},n\in\mathbb{Z}$