Question

sketch the graph of the following function and determine whether the function has any absolute extreme values on its domain. explain how your answer is c with the extreme value theorem.
$y = 4sin x$, $0 < x < 2pi$
determine whether the function has any absolute extreme values on its domain. choose the correct option below and fill in the input boxes as needed (type an exact answer, using $pi$ as needed.)
a. the function has an absolute maximum value at $x = square$ and an absolute minimum value at $x = square$ on its domain.
b. the function has an absolute minimum value at $x = square$ but does not have an absolute maximum value on its domain.
c. the function has an absolute maximum value at $x = square$ but does not have an absolute minimum value on its domain.
d. the function does not have any absolute extreme values on its domain.

Explanation:

Step1: Analyze the function \(y = 4\sin x\)

The range of \(\sin x\) is \([- 1,1]\). So the range of \(y = 4\sin x\) is \([-4,4]\). But the domain is \(0\lt x\lt2\pi\).

Step2: Check the Extreme Value Theorem conditions

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 an absolute maximum and an absolute minimum on \([a,b]\). Here, the domain \(0\lt x\lt2\pi\) is an open interval.
For \(y = 4\sin x\), when \(x\) approaches \(0^{+}\), \(y = 4\sin x\) approaches \(0\); when \(x\) approaches \((2\pi)^{-}\), \(y = 4\sin x\) approaches \(0\). The function \(y = 4\sin x\) is continuous on \((0,2\pi)\) (since \(\sin x\) is continuous everywhere and a constant multiple of a continuous function is continuous), but because the domain is open, the function does not attain the values \(y = 4\) (which would occur at \(x=\frac{\pi}{2}\) if \(x=\frac{\pi}{2}\) was included in a closed - interval extension) and \(y=-4\) (which would occur at \(x = \frac{3\pi}{2}\) if \(x=\frac{3\pi}{2}\) was included in a closed - interval extension) within the open interval \((0,2\pi)\).

Answer:

D. The function does not have any absolute extreme values on its domain.