Question

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

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