Expert: Paul Klarreich - 2/2/2008

Question
Given the following conditions, sketch the given funcion
a.) f(0)=f(2)=0
b.) f'(x)<0 if x<1
c.) f'(1)=0
d.) f'(x)>0 if x>1
e.)f''(x)<0
i can'y figure out parts c and e. i know that in part a it gives two coordinates (0,0) and (2,0) and that part be means there is an increase in slope at x is greater than 1 and that part d means that there is a decrease in slope a x is less than 1. can you help me?

Answer
Questioner:   Julianne
Category:  Calculus
Private:  No
 
Subject:  calculus
Question:  Given the following conditions, sketch the given function
a.) f(0)=f(2)=0
b.) f'(x)<0 if x<1
c.) f'(1)=0
d.) f'(x)>0 if x>1
e.)f''(x)<0
i can'y figure out parts c and e. i know that in part a it gives two coordinates (0,0) and (2,0) and that part be means there is an increase in slope at x is greater than 1 and that part d means that there is a decrease in slope a x is less than 1. can you help me?
.........................................
Hi, Julianne,

[Please, in the future, PROOFREAD your submission and write it as if you were sending it as part of your resume.]

Subject:  calculus
Question:  Given the following conditions, sketch the given function
a.) f(0)=f(2)=0
b.) f'(x)<0 if x<1
c.) f'(1)=0
d.) f'(x)>0 if x>1
e.)f''(x)<0

I can't figure out parts c and e. I know that in part a it gives two coordinates [TWO POINTS WHOSE COORDINATES ARE (0,0) and (2,0) and that part be means there is an increase in slope at x is greater than 1 and that part d means that there is a decrease in slope a x is less than 1. can you help me?
 
Here are the conclusions you will make, then you will draw a picture that is consistent with the conclusions.

a.) f(0)=f(2)=0
The graph passes through  (0,0) and (2,0).  Mark those points.

b.) f'(x)<0 if x<1
To the left of x = 1, the graph is falling.  Make sure it goes down until it reaches x = 1.

c.) f'(1)=0

The graph has a horizontal tangent at x = 1. Probably min or max.  [But you said it goes down until x = 1, didn't you?  So it won't be a max.]

d.) f'(x)>0 if x>1

To the right of x = 1, the graph is rising.  Make sure it goes upward after it leaves x = 1.  That surely makes  x = 1 a minimum, doesn't it?

e.)f''(x)<0

The graph is always concave down. That's going to be hard, and it is not consistent with the other facts.  If  f''(x)  < 0 [is negative], then f' is decreasing.  But f' is negative on the left, then positive, so it has to INCREASE.

Check the problem again.