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.)
The First-Derivative Test is a mathematical tool used to determine whether a critical point on a function is a relative maximum or minimum. It involves taking the first derivative of the function and evaluating it at the critical point.
To find the critical points using the First-Derivative Test, you first take the derivative of the function and set it equal to zero. Solve for the variable to find the x-values of the critical points. Then, plug these values into the first derivative to determine if they are relative maximums or minimums.
No, the First-Derivative Test can only be used for differentiable functions. This means that the function must have a defined derivative at every point on its domain.
A relative maximum is the highest point on a function within a specific interval, while a relative minimum is the lowest point within that interval. These points can also be referred to as local maximums and minimums.
One limitation of the First-Derivative Test is that it can only determine relative maximums and minimums, not absolute maximums and minimums. Additionally, it may not work for functions with discontinuities or sharp corners.