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

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.\n$y = 4sin x$, $0 < x < 2pi$\ndetermine 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.)\na. the function has an absolute maximum value at $x = square$ and an absolute minimum value at $x = square$ on its domain.\nb. the function has an absolute minimum value at $x = square$ but does not have an absolute maximum value on its domain.\nc. the function has an absolute maximum value at $x = square$ but does not have an absolute minimum value on its domain.\nd. the function does not have any absolute extreme values on its domain.

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.\n$y = 4sin x$, $0 < x < 2pi$\ndetermine 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.)\na. the function has an absolute maximum value at $x = square$ and an absolute minimum value at $x = square$ on its domain.\nb. the function has an absolute minimum value at $x = square$ but does not have an absolute maximum value on its domain.\nc. the function has an absolute maximum value at $x = square$ but does not have an absolute minimum value on its domain.\nd. the function does not have any absolute extreme values on its domain.

Answer

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.