Expert: Paul Klarreich - 12/7/2007

Question
Dear Mr. Klarreich,
You helped me a great deal on two questions that I recently asked, and was hoping that you would not mind helping me on another.

I was given a graph of the derivative of f, and told that "the domain of f is the set of all real numbers x such that -10 is less than or equal to x, which is less than or equal to 10.
a) For what values of x does the graph of f have a horizontal tangent?
b)For what values of x in the interval (-10,10) does f have a relative maximum? Justify your answer.
c) For what values of x is the graph of f concave downward?
Thanks again!

Answer
Questioner:   Patsy
Category:  Advanced Math
Private:  No
 
Subject:  calculus
Question:  Dear Mr. Klarreich,
You helped me a great deal on two questions that I recently asked, and was hoping that you would not mind helping me on another.

I was given a graph of the derivative of f, and told that

>>Well, it would be nice to see it, but since you are not asking me about your cat, the site won't let you send me the picture. [Complain to them, please.  They don't listen to me, maybe they will listen to you.  Tell them you'll be happy to click on their advertising links.]

the domain of f is -10 <= x <=10.
a) For what values of x does the graph of f have a horizontal tangent?

Look for places where the graph touches or crosses the x-axis.  That makes f'(x) = 0 and gives you the H.T.

c) For what values of x is the graph of f concave downward?

I know, b comes before c, but:

concave down ==> f'' is neg ==> f' is decreasing ==> graph of f' is falling.
concave up   ==> f'' is pos ==> f' is increasing ==> graph of f' is rising.


b)For what values of x in the interval (-10,10) does f have a relative maximum? Justify your answer.

f has a relative max ==> f' = 0 AND f'' is negative.  Combine the answers to a,c:

Find x where the graph of f' touches the x-axis AND is falling.