a. for what values of x does f(x) = 9x - 9 sin x have a horizontal tangent line?\nb. for what values of x…

a. for what values of x does f(x) = 9x - 9 sin x have a horizontal tangent line?\nb. for what values of x does f(x) = 9x - 9 sin x have a slope of 9?\na. choose the correct answer below.\na. x = 2π\nb. x = 2π+2kπ, where k is any integer\nc. x = 2π + kπ, where k is any integer\nd. there are no values of x where f(x) = 9x - 9 sin x has a horizontal tangent line

a. for what values of x does f(x) = 9x - 9 sin x have a horizontal tangent line?\nb. for what values of x does f(x) = 9x - 9 sin x have a slope of 9?\na. choose the correct answer below.\na. x = 2π\nb. x = 2π+2kπ, where k is any integer\nc. x = 2π + kπ, where k is any integer\nd. there are no values of x where f(x) = 9x - 9 sin x has a horizontal tangent line

Answer

Explanation:

Step1: Find the derivative of f(x)

Given (f(x)=9x - 9\sin x), using the sum - difference rule and derivative formulas ((x^n)^\prime=nx^{n - 1}) and ((\sin x)^\prime=\cos x), we have (f^\prime(x)=(9x)^\prime-(9\sin x)^\prime=9 - 9\cos x).

Step2: Solve for horizontal tangent (slope = 0)

A horizontal tangent line has a slope of 0. Set (f^\prime(x)=0), so (9 - 9\cos x = 0). First, factor out 9: (9(1-\cos x)=0), then (1-\cos x = 0), which gives (\cos x=1). The solutions of (\cos x = 1) are (x = 2k\pi), where (k) is any integer.

Step3: Solve for slope = 9

Set (f^\prime(x)=9), so (9 - 9\cos x=9). Subtract 9 from both sides: (-9\cos x=0), then (\cos x = 0). The solutions are (x=\frac{\pi}{2}+k\pi), (k\in\mathbb{Z}).

Answer:

a. B. (x = 2k\pi), where (k) is any integer b. (x=\frac{\pi}{2}+k\pi), where (k) is any integer