(tangent function behavior mc)\nthe function h is defined as h(θ) = tan θ. identify all inflection points on…

Answer:

$\theta=\pi$

Explanation:

Step1: Find the first - derivative

The derivative of $y = \tan\theta$ is $y'=\sec^{2}\theta$.

Step2: Find the second - derivative

Using the chain - rule, if $y'=\sec^{2}\theta$, then $y'' = 2\sec\theta\cdot\sec\theta\tan\theta=2\sec^{2}\theta\tan\theta$.

Step3: Set the second - derivative equal to zero

We set $y'' = 0$, so $2\sec^{2}\theta\tan\theta=0$. Since $\sec^{2}\theta=\frac{1}{\cos^{2}\theta}\neq0$ for all real $\theta$ such that $\cos\theta\neq0$, we solve $\tan\theta = 0$. The general solution of $\tan\theta=0$ is $\theta = n\pi$, where $n\in\mathbb{Z}$.

Step4: Find the solution in the given interval

We are looking for solutions in the interval $\left(\frac{\pi}{2},\frac{3\pi}{2}\right]$. When $n = 1$, $\theta=\pi$ which lies in the interval $\left(\frac{\pi}{2},\frac{3\pi}{2}\right]$.