Question

does the graph of the function below have any horizontal tangent lines in the interval 0 ≤ x ≤ 2π? if so, where? if not, why not? visualize your findings by graphing function with a grapher. y = 7x + 6 sin x b. the function has no horizontal tangent lines in the interval 0 ≤ x ≤ 2π because it is never zero in the interval 0 ≤ x ≤ 2π c. the function has no horizontal tangent lines in the interval 0 ≤ x ≤ 2π because it is defined at every point in the interval 0 ≤ x ≤ 2π d. the function has no horizontal tangent lines in the interval 0 ≤ x ≤ 2π because its derivative is never zero in the interval 0 ≤ x ≤ 2π confirm the result visually by graphing y = 7x + 6 sin x. choose the correct graph below.

Explanation:

Step1: Recall the condition for horizontal tangent

A function $y = f(x)$ has a horizontal tangent when $y'=f'(x)=0$. First, find the derivative of $y = 7x + 6\sin x$.
Using the sum - rule of differentiation and the basic derivative formulas $\frac{d}{dx}(x)=1$ and $\frac{d}{dx}(\sin x)=\cos x$, we have $y'=\frac{d}{dx}(7x + 6\sin x)=7 + 6\cos x$.

Step2: Solve $y' = 0$ for $x$ in the interval $[0,2\pi]$

Set $7 + 6\cos x=0$. Then, $\cos x=-\frac{7}{6}$. But the range of the cosine function is $[- 1,1]$, and $-\frac{7}{6}
otin[-1,1]$. So, there is no $x\in[0,2\pi]$ such that $y' = 0$. This means the function has no horizontal tangent lines in the interval $0\leq x\leq2\pi$ because its derivative is never zero in the interval $0\leq x\leq2\pi$.

Answer:

D. The function has no horizontal tangent lines in the interval $0\leq x\leq2\pi$ because its derivative is never zero in the interval $0\leq x\leq2\pi$