Answer
No horizontal tangential line exists for this function.
Work Step by Step
Horizontal lines have, by definition, a slope of $m = 0$. Therefore, for this exercise, we are trying to find a value of $x$ where $f'(x) = 0$.
Combining the Addition and Power rules, we can find $f'(x)$ as follows: $$f(x) = 3x + sinx + 2$$ $$f'(x) = 3x^{1-0} + (cosx) + 0$$ $$f'(x) = 3 + cosx$$ where the domain of $x$ is $[0, 2\pi]$. Solving for $f'(x) = 0$: $$0 = 3 + cosx$$ $$-3 = cosx$$
Since there is no solution to this equation within the domain of $x$ $[0, 2\pi]$, it can be concluded that no horizontal tangential line exists for this function.
