Question

find the absolute maximum and minimum values of the following function on the given interval. then graph the function. identify the points on the graph where the absolute extrema occur.

$f(\theta)=\cos\theta,\frac{2\pi}{3}\leq\theta\leq2\pi$

find the absolute maximum. select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.

a. the absolute maximum value occurs at $\theta=$
(use a comma to separate answers as needed. type exact answers, using $\pi$ as needed.)

b. there is no absolute maximum

Explanation:

Step1: Find the derivative of the function

The derivative of \(f(\theta)=\cos\theta\) is \(f'(\theta)=-\sin\theta\).
Set \(f'(\theta) = 0\), so \(-\sin\theta=0\), which gives \(\theta = k\pi,k\in\mathbb{Z}\).
In the interval \(\frac{2\pi}{3}\leq\theta\leq2\pi\), the critical points are \(\theta=\pi\) and \(\theta = 2\pi\).

Step2: Evaluate the function at critical points and endpoints

• Evaluate \(f(\theta)\) at \(\theta=\frac{2\pi}{3}\): \(f(\frac{2\pi}{3})=\cos(\frac{2\pi}{3})=-\frac{1}{2}\)
• Evaluate \(f(\theta)\) at \(\theta=\pi\): \(f(\pi)=\cos(\pi)= - 1\)
• Evaluate \(f(\theta)\) at \(\theta = 2\pi\): \(f(2\pi)=\cos(2\pi)=1\)

Answer:

A. The absolute maximum value \(1\) occurs at \(\theta = 2\pi\)