Find all critical numbers of of the function $f ( \theta ) = 2 \sec \theta + \tan \theta , 14 \pi

Question

Find all critical numbers of of the function $f ( \theta ) = 2 \sec \theta + \tan \theta , 14 \pi < \theta < 16 \pi .$
A) $\theta = \frac { 91 \pi } { 6 }$
B) $\theta = \frac { 91 \pi } { 6 } , \frac { 95 \pi } { 6 }$
C) $\theta = \frac { 89 \pi } { 6 } , \frac { 95 \pi } { 6 }$
D) $\theta = \frac { 95 \pi } { 6 }$
E) $\theta = \frac { 31 \pi } { 2 }$

Quizplus · Accepted Answer

To find the critical numbers of $f(\theta) = 2\sec\theta + \tan\theta$, we first find its derivative: $f'(\theta) = 2\sec\theta\tan\theta + \sec^2\theta$. Setting the derivative equal to zero gives us the critical points, but we must also consider where the derivative is undefined due to the nature of the secant and tangent functions. The critical points occur where $f'(\theta) = 0$ and where $f'(\theta)$ is undefined, which happens when $\theta = \frac{(2n+1)\pi}{2}$ for any integer $n$, because secant and tangent are undefined for these values. Within the given interval $14\pi < \theta < 16\pi$, the critical points that satisfy these conditions are $\theta = \frac{91\pi}{6}$ and $\theta = \frac{95\pi}{6}$, corresponding to where the derivative is zero or the function's behavior changes due to the periodic nature of secant and tangent.

Find All Critical Numbers of Of the Function f(θ)=2sec⁡θ+tan⁡θ,14π<θ<16π.f ( \theta ) = 2 \sec \theta + \tan \theta , 14 \pi < \theta < 16 \pi .f(θ)=2secθ+tanθ,14π<θ<16π.

Find All Critical Numbers of Of the Function $f ( \theta ) = 2 \sec \theta + \tan \theta , 14 \pi < \theta < 16 \pi .$