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posted by  Aerowen on 11/6/2009 3:28:25 PM  |  status: Closed  |  Earned Karma: 35

Curve Sketching -- monotonic functions and inflection points

Course Textbook Chapter Problem Needs by
Calculus Calculus: Early Transcendentals (6th) by Stewart 4.5-4.6 N/A 11/6/2009 at 10:00:00 PM
Question Details:
This is the exercise I'm having trouble with:
Let f be defined by f(x) = x4ln(x).
a) Find a number a so that f is monotone on each of the intervals (0, a) and (a, ∞). Determine if f is increasing or decreasing on each interval.
a = ?
increasing/decreasing on (0,a)
increasing/decreasing on (a, ∞)
b) Find a number b so that f is concave in one direction on each of the intervals (0, b) and (b, ∞). Determine if f is concave up or down on each interval.
b = ?
concave up/concave down on (0, b)
concave up/concave down on (b, ∞).
-------------------------------------------------
I'm pretty sure the first part is the first derivative, and the second part is the second derivative, since we are talking about monotonic and inflection... but I just can't get them. Any help would be greatly appreciated!
"You cannot depend on your eyes when your imagination is out of focus." - Mark Twain
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Oracle
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(Drexel University)
posted by The Big Unit on 11/6/2009 3:38:10 PM  |  status: Live
Asker's Rating: Lifesaver   
Response Details:
Here are some helpful graphs of the given function:
a) You have the function . Taking the first derivative yields:
From this, it follows that:
 if
 if
From this, you can conclude that:
 is decreasing on
 is increasing on
So your number  will be  (ANSWER 1)
b) Taking the second derivative yields:
Because  when  and  when , this means that  will be concave down on  and concave up on .
So your number  will be  (ANSWER 2)
Jon AKA THE BIG UNIT
If you have any questions about the problem, just message me and I will help you out. Hope this helps!
 
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