Applications of Differentiation

Question

A function f is continuous on the closed interval [-3, 3] such that f(-3)=4 and f(3)=1. The function f' and f'' have the following properties:

 

  x:    -3<x<-1      x=-1         -1<x<1    x=1      1<x<3

f'(x)   Positive    Undefined      negative    0         negative

f'(x)   Positive    Undefined      Negative    0         Negative

 

1.) Determine the intervals where the graph of f is concave down. Be sure to use interval notation and to justify your answer

 

2.) Using interval notation, determine where f is decreasing on the interval [-3, 3]. Justify your answer.

 

 

Confused on these questions on this problem; could you show your steps on how you got your answer so I understand?

Answer 1

To find the interval where the graph is decreasing or increasing, we set f'(x) equal to zero.  This is called the first derivativ test.  When we solve for x from the derivative, we get our critical points, or locations of maximum and minimum value.  If the first derivative is negative, then f(x) is decreasing.  If the derivative is positive, then f(x) is increasing.

 

 

To find the interval where the graph is concave up, we set f''(x) equal to zero.  This is called the second derivative test.  When we solve for x from this equation, we get our infection points.  This points are the location where the graph changes concavity.  If the second derivative is negative, then it is concave down.  If the second derivative is positive, the f(x) is concave up.

 

Notice that f'(-1) and f''(-1) are undefined.  This should tell you that f(x) is either a rational function or a radical function.

 

Because f'(-1) is undefined, there is no maximum or minimum at x=-1.  In addition, there is no maximum or minimum at x=1 because f'(x) before and after this x value does not change signs.

 

x=-1 is an asymptote, however, there is some action going one around x=-1.  So we keep this for the second derivative.

 

Now that you understand the derivative tests, we use the information given.  I will assume the second row of the information shows f''(x).

 

f'(x) is positive in the interval [-3, -1).  This where f(x) increases.

f'(x) is negative in the interval (-1, 3].  This is where f(x) decreases.

 

 

f''(x) is negative in the interval (-1, 3].   This is where f(x) is concave down.