sketch the graph of the following function and determine whether the function has any absolute extreme…

Explanation:

Step1: Recall properties of sine - function

The general form of a sine - function is $y = A\sin(Bx - C)+D$. For the function $y = 5\sin x$, $A = 5$, $B = 1$, $C = 0$, and $D = 0$. The amplitude of the function $y = A\sin x$ is $|A|$. Here, the amplitude is $|5|=5$, which means the graph of $y = 5\sin x$ oscillates between $y=-5$ and $y = 5$. The period of the function $y=\sin x$ is $T=\frac{2\pi}{B}$. Since $B = 1$, the period of $y = 5\sin x$ is $2\pi$.

Step2: Analyze the domain

The domain is given as $0\lt x\lt2\pi$. When $x = 0$, the function is not defined in our given domain, but $\lim_{x\rightarrow0^{+}}5\sin x=0$, and when $x = 2\pi$, the function is not defined in our given domain, but $\lim_{x\rightarrow2\pi^{-}}5\sin x = 0$.

Step3: Find extreme values

We know that the derivative of $y = 5\sin x$ is $y'=5\cos x$. Set $y' = 0$, then $5\cos x=0$, which gives $\cos x = 0$. In the interval $(0,2\pi)$, $x=\frac{\pi}{2}$ and $x=\frac{3\pi}{2}$. When $x=\frac{\pi}{2}$, $y = 5\sin\frac{\pi}{2}=5$; when $x=\frac{3\pi}{2}$, $y = 5\sin\frac{3\pi}{2}=-5$.

Step4: Apply the extreme - value theorem

The extreme - value theorem states that if a function $y = f(x)$ is continuous on a closed interval $[a,b]$, then $f(x)$ has both a maximum and a minimum value on $[a,b]$. Our function $y = 5\sin x$ is continuous on $(0,2\pi)$ but the interval is open. So, although the function has local maximum value of $5$ at $x=\frac{\pi}{2}$ and local minimum value of $-5$ at $x=\frac{3\pi}{2}$ in the open interval $(0,2\pi)$, it does not have absolute extreme values on the open interval $(0,2\pi)$ because the endpoints are not included.

Step5: Sketch the graph

The graph of $y = 5\sin x$ starts just above the origin (since $x\gt0$), goes up to $y = 5$ at $x=\frac{\pi}{2}$, comes back down to $y = 0$ at $x=\pi$, goes down to $y=-5$ at $x=\frac{3\pi}{2}$, and then comes back up to just below $y = 0$ as $x$ approaches $2\pi$.

Answer:

The function $y = 5\sin x$ on the domain $0\lt x\lt2\pi$ has no absolute extreme values. The graph of $y = 5\sin x$ is a sine - wave with amplitude $5$ and period $2\pi$ on the open interval $(0,2\pi)$.