determine whether rolles theorem applies to the given function on the given interval. if so, find the…
determine whether rolles theorem applies to the given function on the given interval. if so, find the point(s) that are guaranteed to exist by rolles theorem. f(x) = cos 2x; π/4, 3π/4 select the correct choice and, if necessary, fill in the answer box to complete your choice. a. rolles theorem applies and the point(s) guaranteed to exist is/are c = (type exact answer(s). use a comma to separate answers as needed.) b. rolles theorem does not apply.
Answer
Explanation:
Step1: Check continuity
The function $y = \cos(2x)$ is a cosine - type function. Cosine functions are continuous everywhere. So, $f(x)=\cos(2x)$ is continuous on the closed interval $\left[\frac{\pi}{4},\frac{3\pi}{4}\right]$.
Step2: Check differentiability
The derivative of $y = \cos(2x)$ using the chain - rule is $f^\prime(x)=-2\sin(2x)$. Sine functions are differentiable everywhere. So, $f(x)$ is differentiable on the open interval $\left(\frac{\pi}{4},\frac{3\pi}{4}\right)$.
Step3: Check $f(a)=f(b)$
When $x = \frac{\pi}{4}$, $f\left(\frac{\pi}{4}\right)=\cos\left(2\times\frac{\pi}{4}\right)=\cos\left(\frac{\pi}{2}\right)=0$. When $x=\frac{3\pi}{4}$, $f\left(\frac{3\pi}{4}\right)=\cos\left(2\times\frac{3\pi}{4}\right)=\cos\left(\frac{3\pi}{2}\right)=0$. Since $f\left(\frac{\pi}{4}\right)=f\left(\frac{3\pi}{4}\right)$, Rolle's Theorem applies.
Step4: Find $c$
Set $f^\prime(c) = 0$. Since $f^\prime(x)=-2\sin(2x)$, then $-2\sin(2c)=0$, which implies $\sin(2c)=0$. We know that $\sin\theta = 0$ when $\theta = k\pi,k\in\mathbb{Z}$. So, $2c = k\pi$, or $c=\frac{k\pi}{2}$. We want $c\in\left(\frac{\pi}{4},\frac{3\pi}{4}\right)$. When $k = 1$, $c=\frac{\pi}{2}$.
Answer:
A. Rolle's Theorem applies and the point(s) guaranteed to exist is/are $c=\frac{\pi}{2}$