The discussion revolves around the application of the first-derivative test to find absolute extrema of a function, specifically at the points x = -8.8 and x = -7.2. Participants are analyzing the behavior of the derivative g'(x) and its implications for identifying maximum and minimum values.
There is an ongoing exploration of the implications of the derivative's sign changes and the presence of a vertical asymptote at x = -8. Some participants have offered insights into the behavior of g'(x) around the critical points, while others are clarifying their understanding of the terminology used in the context of extrema.
Participants are working with a specific function and its derivative, with a focus on critical points and the effects of asymptotes. There is an acknowledgment of potential misunderstandings regarding the signs of the derivative in different intervals.
MathDude said:-8.8 appears to be a relative min if g'(x) changes from - to +.
Think about it down and then up: \ /
seems like a minimum
MathDude said:I had a great right up for you, but when I posted it I was timed out and had to login in (drives me nuts!) and lost everything. I don't have time to write it again, but here is a short version.
g'(x) is
+ (-inf, -8.8)
- (-8.8, -8)
- (-8, -7.2) Asymptote at -8
+ (-7.2, + inf)
I didn't observe the vertical asymptote at -8 in my earlier post. This changes the slope.
-8.8 is actually a max
-7.2 is a min.
Relative implies either local or abs. Evaluating g(x) at the critical points will enable you to tell if its local or abs. Abs values will yields the very largest and very smallest values for g(x) for all values of x within the given domain. Locals give you min or max within a local range. Locals can be absolute if there is only one max and/or one min. Say you have two rel max, the point with a larger g(x) will be the abs and the smaller g(x) will be the local.
Mark44 said:g'(x) can change signs only at zeros in the numerator, which are x = -8.8 and x = -7.2. For x < -8.8, g'(x) > 0. For x > -7.2, g'(x) > 0. For -8.8 < x < -7.2, g'(x) < 0 (although g'(x) is undefined at x = -8.)