Question

question 17 · 1 point find the critical points of the function f(x)= - 5cos²(x) contained in the interval (0,2π). use a comma to separate multiple critical points. enter an exact answer. provide your answer below: x =

Explanation:

Step1: Find the derivative of the function

We use the chain - rule. If $y = - 5\cos^{2}(x)$, let $u=\cos(x)$, then $y=-5u^{2}$. The derivative of $y$ with respect to $u$ is $y^\prime_{u}=-10u$, and the derivative of $u$ with respect to $x$ is $u^\prime_{x}=-\sin(x)$. By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$, so $f^\prime(x)=-10\cos(x)(-\sin(x)) = 5\sin(2x)$ (using the double - angle formula $\sin(2x) = 2\sin(x)\cos(x)$).

Step2: Set the derivative equal to zero

Set $f^\prime(x)=0$, so $5\sin(2x)=0$. Then $\sin(2x)=0$.

Step3: Solve for $x$ in the given interval

If $\sin(2x)=0$, then $2x = n\pi$, where $n\in\mathbb{Z}$. So $x=\frac{n\pi}{2}$. For the interval $(0,2\pi)$, when $n = 1,x=\frac{\pi}{2}$; when $n = 2,x=\pi$; when $n = 3,x=\frac{3\pi}{2}$.

Answer:

$\frac{\pi}{2},\pi,\frac{3\pi}{2}$