Question

Determine the point(s) (if any) at which the graph of the function has a horizontal tangent line. $$ y=x+\sin x, \quad 0 \leq x<2 \pi $$

Short answer

The point at which the graph of the function has a horizontal tangent line is \( (\pi , \pi) \).

Step-by-step solution

Find the Derivative of the Function

To find where a function has a horizontal tangent line, we first need to find the derivative of the function. \n The derivative of \( y = x + \sin{x} \) with respect to \( x \) is \( y' = 1 + \cos{x} \) using the chain rule.

Set Derivative Equal to 0

After obtaining the derivative, the next step is to set it equal to 0 and solve for \( x \) since we are interested in finding when the slope (rate of change) is 0. This results in the equation \( 1 + \cos{x} = 0 \). Solving this gives us \( \cos{x} = -1 \). Recall that cosine function equals -1 at \( x = \pi \).

Find the Corresponding y Values

The last step requires plugging in the obtained \( x \) value into the original function to find the corresponding \( y \) value(s). When \( x = \pi \), \( y = x + \sin{x} = \pi + \sin{\pi} = \pi + 0 = \pi \). Hence, the coordinates of the point where the function has a horizontal tangent line is \( (\pi , \pi) \).