list the critical numbers of the following function in increasing order. enter n in any blank that you dont…

Explanation:

Step1: Find the derivative

Using the chain - rule and derivative formulas. The derivative of $\cos(\theta)$ is $-\sin(\theta)$ and the derivative of $\sin^{2}(\theta)$ is $2\sin(\theta)\cos(\theta)$. $f'(\theta)=-6\sin(\theta)+6\sin(\theta)\cos(\theta)$

Step2: Factor the derivative

Factor out $6\sin(\theta)$ from $f'(\theta)$. $f'(\theta)=6\sin(\theta)(\cos(\theta) - 1)$

Step3: Set the derivative equal to zero

Set $f'(\theta)=0$, then we have two cases: Case 1: $6\sin(\theta)=0$, which gives $\sin(\theta)=0$. In the interval $-\pi\leq\theta\leq\pi$, $\theta =-\pi,0,\pi$. Case 2: $\cos(\theta)-1 = 0$, which gives $\cos(\theta)=1$. In the interval $-\pi\leq\theta\leq\pi$, $\theta = 0$.

Answer:

$-\pi$ $0$ $\pi$