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 the function with a grapher. y = 7x + 6 sin x select the correct answer below and, if necessary, fill in the answer box to complete your choice. a. the function has a horizontal tangent line at x = (type your answer in radians. type an exact answer in terms of π. use a comma to separate answers as needed.) 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

Explanation:

Step1: Find the derivative of the function

The derivative of $y = 7x+6\sin x$ using the sum - rule and basic derivative formulas. The derivative of $7x$ is $7$ and the derivative of $6\sin x$ is $6\cos x$. So, $y'=7 + 6\cos x$.

Step2: Set the derivative equal to zero

We want to find where $y'=0$, so we set up the equation $7 + 6\cos x=0$. Then, we solve for $\cos x$:
\[

$$\begin{align*} 6\cos x&=- 7\\ \cos x&=-\frac{7}{6} \end{align*}$$

\]

Step3: Analyze the range of the cosine function

The range of the cosine function is $[-1,1]$. Since $-\frac{7}{6}\approx - 1.17
otin[-1,1]$, there are no solutions for $x$ in the real - number system. In the interval $0\leq x\leq2\pi$, the derivative $y' = 7 + 6\cos x$ is never zero.

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 $0\leq x\leq2\pi$