What does f ' and f '' say about f ?

Erin Horst

Now that we have explored derivatives, we can begin to assess what f' and f'' say about f. You may already have recognized some properties of first and second derivatives, but here we will lay them all out.

The following statements are true:

a.) If f'(x) >0 on an interval, then f is increasing on that interval.

b.) If f'(x) <0 on an interval, then f is decreasing on that interval.

c.) If f''(x) >0 on an interval, then f is concave upward on that interval

d.) If f''(x) <0 on an interval, then f is concave downward on that interval.

e.) If f'(x)=0, then the x value is a point of inflection for f.


To illustrate these principles, consider the following problems.

1.) Suppose

a.) On what interval is f increasing? On what interval is f decreasing?

b.) Does f have a maximum or minimum value?


2.) Sketch the graph of a function whose first and second derivatives are always negative.


3.) Sketch the graph of the function that satisfies the given conditions.

4.) The cost of living continues to rise, but at a slower rate. In terms of function and its dertivatives, what does this statement mean?


5.) The president annouces that the national deficit is increasing, but at a decreasing rate. Interpret this statement in terms of a function and its derivatives.


The above problems were adapted from Stewart, J. (1998). Calculus: Concepts and contexts. Pacific Grove, CA: Brooks/Cole Publishing Co.