© 2014 Shmoop University, Inc. All rights reserved.
 

Topics

Introduction to Second Derivatives And Beyond - At A Glance:

We say a function f is concave down if it curves downward like an upside-down spoon (concave side down):

It's also fine to have only part of the bowl. Both of these functions are concave down:

"f is concave down" means exactly the same thing as "' is decreasing" or "the slope of f is decreasing." If we have a bowl, then ' goes from positive to zero to negative, so ' is decreasing:

If f is increasing and concave down, then the slope of f starts positive and decreases—in other words, ' is decreasing:

If f is decreasing and concave down, then the slope of f starts negative and becomes steeper (more negative)—in other words, ' is decreasing:

Saying that a differentiable function is decreasing is the same as saying the derivative of that function is negative. Assuming that ' is differentiable, saying that ' is decreasing is the same as saying " is negative. So the following statements all mean the same thing:

  • f is concave down.
      
  • ' is decreasing.
      
  • " is negative.

Exercise 1

Determine if the statement is equivalent to (mean the same thing as) the statement

"f is concave down."

Explain your answer (whether the statement is equivalent or not).

f is decreasing.

Exercise 2

Determine if the statement is equivalent to (mean the same thing as) the statement

"f is concave down."

Explain your answer (whether the statement is equivalent or not).

f is negative.

Exercise 3

Determine if the statement is equivalent to (mean the same thing as) the statement

"f is concave down."

Explain your answer (whether the statement is equivalent or not).

' is decreasing.

Exercise 4

Determine if the statement is equivalent to (mean the same thing as) the statement

"f is concave down."

Explain your answer (whether the statement is equivalent or not).

' is negative.

Exercise 5

Determine if the statement is equivalent to (mean the same thing as) the statement

"f is concave down."

Explain your answer (whether the statement is equivalent or not).

" is decreasing.

Exercise 6

Determine if the statement is equivalent to (mean the same thing as) the statement

"f is concave down."

Explain your answer (whether the statement is equivalent or not).

" is negative.

Advertisement
Advertisement
Advertisement
back to top