Originally posted by davefoc
1. Does on [a,b] mean in the interval between a & b?
Yes. More precisely, it's the closed interval from
a to
b. That means it includes the endpoints
a and
b. The notation is standard. The notation for open intervals, i.e., those that don't include their endpoints, uses parentheses instead of square brackets, like this: (
a,
b). You can also use one square bracket and one parenthesis, e.g., [
a,
b) or (
a,
b], to denote a half-open interval, i.e., an interval that includes one endpoint and excludes the other.
2. If the concavity changes is that by definition an inflection point?
Yes, at least according to the two calculus books I just checked.
3. If the concavity changes does that preclude the possibility that it's a relative minimum?
I'm pretty sure that's correct, unless
f doesn't have a first derivative at
x = 0, in which case a relative minimum is definitely possible. The question allows that case, as it never states that 0 is in [
a,
b], though the omission was probably an oversight.
4. Does the reference to the first derivative test mean to find out if the first derivative at the point in question is equal to zero in which case we will be either at a local minimum or a local maximum?
I don't understand the reference to the first derivative test. For that matter, I don't understand the question, "What test do we apply here?". My first reaction was, "Well, that depends. What do you want to know?"
A zero first derivative at a point doesn't mean the point is a local maximum or minimum. However, at any local maximum or minimum, the derivative will be zero if it exists at all, which it needn't.
Mathworld has a
page about the first derivative test.
