Expert: Paul Klarreich - 3/23/2009

Question
Will you please help me figure out these things I cant figure them out.  Thank you so much.

given the equation f(x)= 9*Sqrt(x)*e^(-x)

a) Find the interval on which f is increasing.
b) Find the interval on which f is decreasing.
c) Find the local maximum value of f.
D) Find the inflection point.
e) Find the interval on which f is concave up.
f) Find the interval on which f is concave down.


Answer
Questioner:   cornelius
Country:  United States
Category:  Calculus
Private:  No
 
Subject:  Calculus how derivatives affect the shape of a graph
Question:  Will you please help me figure out these things I cant figure them out.  Thank you so much.

given the equation f(x)= 9*Sqrt(x)*e^(-x)

>> It is not clear WHY you can't figure them out, so I will get you started.  If you get stuck, tell me what you did and I'll see If I can help.

If

f(x) = Sqrt(x) e^(-x)   << the 9 is a waste of time.

Find f'(x) using the product rule:

         e^(-x)
f'(x) = --------- - sqrt(x) e^(-x)
        2 sqrt(x)

Do a little algebra:

              1 - 2x
f'(x) = e^(-x) -------
              sqrt(x)

Now keep in mind that:
1. e^(anything) is positive,
2. sqrt(x) is positive
and you will have little trouble with these:


a) Find the interval on which f is increasing.

>> find where  f' is positive. That basically means 1-2x > 0

.......................
b) Find the interval on which f is decreasing.

>> find where  f' is negative. That basically means 1-2x < 0

.........................
c) Find the local maximum value of f.

>> find where  f' is zero. That basically means 1-2x = 0

..........................
D) Find the inflection point.

You will have to find  f''(x).  [Sorry,but that will be some work for you, with both quotient and product rules.]


Then set that f''(x) = 0.

e) Find the interval on which f is concave up.

>> find where  f'' is positive.

.....................................
f) Find the interval on which f is concave down.

>> find where  f'' is negative.

..........................................