Expert: Paul Klarreich - 2/22/2007

Question
Hey Paul!  I believe you answered my snowball question.  Thank you very much.  I have just one more calculus question.  I have found the arc for which f(x)=(x^2-2)*e^(2x) is decreasing to be (-2,1) and the point (1,f(1)) is the graph's absolute minimum.  The question is, does this decreasing arc reach a local (relative) or an absolute minimum?  My first guess is to say local, at least with respect to the whole graph, because the absolute minimum is outside of the decreasing arc.  But I also think that as close as you can get to f(1) will be the arc's absolute minimum.  I'm really not sure if I'm even interpreting the question right.  Hopefully this question is much simpler to answer.  Thanks,
Andy

Answer
Questioner:   andy
Category:  Calculus

Subject:  decreasing INTERVAL
Question:  Hey Paul!  I believe you answered my snowball question.  Thank you very much.  I have just one more calculus question.  

I have found the arc for which f(x)=(x^2-2)*e^(2x) is decreasing to be (-2,1) and the point (1,f(1)) is the graph's absolute minimum.  

>> That's generally called the INTERVAL.

The question is, does this decreasing arc reach a local (relative) or an absolute minimum?  My first guess is to say local, at least with respect to the whole graph, because the absolute minimum is outside of the decreasing arc.  But I also think that as close as you can get to f(1) will be the arc's absolute minimum.  I'm really not sure if I'm even interpreting the question right.  Hopefully this question is much simpler to answer.  Thanks,
Andy
....................................
Hi, Andy,

The value AT x = 1 is indeed the relative minimum.  You will usually find such a point where  f'(x) = 0, and it separates a decreasing interval (f' < 0) on the left from an increasing interval (f' > 0) on the right.

You generally do these problems by:

A. finding f'(x).
B. Locating critical points, which are values of x where:
   1. f' = 0, called stationary points,
   2. f' is undefined, called singular points.
   3. endpoints of the domain of the function, called ... er.... end points.
C.  Separating the domain into subintervals bounded by these critical points, and in each one, determine whether the graph is rising or falling.

Basically, I think you did OK on this one, but don't be afraid to say that f(1) is the minimum.