Chapter 4 - Section 4.1 - Extreme Values of Functions - Exercises - Page 215: 19

The function has an absolute maximum value at $x=\pi/2$, where $y=3$ and an absolute minimum value at $x=3\pi/2$, where $y=-3$.

Work Step by Step

$y=3\sin x$ for $0\lt x\lt2\pi$ The graph is sketched below. - The graph has an absolute maximum value at $x=\pi/2$, where $y=3$. - The graph has an absolute minimum value at $x=3\pi/2$, where $y=-3$ Here we find that even though the domain is an open interval, there exist both absolute maximum and minimum values for the function. This, nevertheless, does not contradict Theorem 1, as Theorem 1 only concerns itself with the function on a CLOSED interval; it does not specify anything when we are talking about an OPEN interval. In fact, Theorem 1 does not mean there cannot be any absolute extreme values for the function on an open interval, which are completely possible as we see in this case.

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