Relative Maxima/Minima of polynomial functions

Question

I am taking the Pre Calculus 12 course online. I came across this concept that the online material teaches in 3 different ways, and each one contradicts the other. I find this extremely frustrating.

Instructor 1 describes Relative maxima and minima as:

The highest or lowest point in the turning point of a function. He specifically clarifies that an absolute minimum is NOT a relative minimum, and vice versa He also states that functions will have several maxima and minima (NOT INCLUDING ABSOLUTE maxima/minima)

Instructor 2 describes them in this way:

He indicates that the absolute maximum and minimum of a function are actually the relative maximum and minimum. His solutions imply that there is only 1 relative max/min, because he ignores the other turning points, and these are actually the absolute max/min, which directly contradicts instructor 1, who states that absolute max/min are not relative max/min

On a practice test, the solution implies that there are multiple maxima and minima, and that the absolute maximum is also a relative maximum, and vice versa.

Essentially I am being taught the same concept three different ways...each of which could interpret the others as incorrect.

There are multiple relative maxima/minima, they do not include the absolute max/min.
There is only one relative max and min; they are the absolute max/min
There are multiple relative maxima/minima; they include the absolute max/min.

...which is correct?

Thank you

@AndréNicolas: Does the exception in (2) apply to all common definitions? — dfeuer, Jul 17 '13 at 02:24
No, it doesn't. I am away from the office, so cannot check with a few standard calculus books. But my memory is that usually endpoint max/min are not classified as local in first-year calculus. — André Nicolas, Jul 17 '13 at 02:39

score 1 · Answer 1 · edited Apr 25 '15 at 16:20

There can be small differences of terminology, but nothing as radical as what you are experiencing.

The function $f(x)$ has a relative maximum (also known as a local maximum) at $x=c$ if there is a positive $\epsilon$ such that $f(x)\le f(c)$ for all $x$ in the interval $(c-\epsilon, c +\epsilon)$.

Sometimes an exception is made when we are maximizing a function over a closed interval $[a,b]$. If there is an $\epsilon$ such that $f(x)\le f(a)$ for all $x$ in the interval $[a,a+\epsilon)$, then some people may say there is a relative maximum at $x=a$. I believe that most standard calculus books do not count endpoint maxima as relative maxima. Do check your text. Whatever it says is, for your purposes, the local dialect. That dialect may change a little when you get to university.

A relative maximum can be an absolute maximum. An absolute maximum can be a relative maximum. If we have an absolute maximum at $x=c$, and our function is defined in some interval $(c-\epsilon, c+\epsilon)$, then the absolute maximimum is automatically a relative maximum.

From your description, the exam is closest to using the words in the standard way. The text may also be substantially correct.

Remark: I am sorry that you are getting mixed messages. The online courses, at least in the Dogwood province, can be on the weak side.

I wonder if we can confirm that there is any text for this online course? I suspect that may be contributing to the lack of stable definitions here. — Daniel R. Collins, Dec 29 '15 at 08:40

Relative Maxima/Minima of polynomial functions

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