4. consider the graph of (y = csc x). its first local minimum for (xgeq0) is at ((\frac{pi}{2},1)). by…

Answer:

$\left(\frac{\pi}{2},5\right)$

Explanation:

Step1: Recall csc - function transformation

The function $y = 5\csc x$ is a vertical - stretch of $y=\csc x$ by a factor of 5.

Step2: Transform the minimum point

For $y = \csc x$, the first local minimum for $x\geq0$ is at $\left(\frac{\pi}{2},1\right)$. For $y = 5\csc x$, the $x$ - coordinate of the minimum point remains the same because there is no horizontal transformation. The $y$ - coordinate is multiplied by 5. So, $y = 5\times1=5$. The new minimum point is $\left(\frac{\pi}{2},5\right)$.