You know that the first derivative,
What does the second derivative,
, tell you about ? What does tell you about ? tells us the same thing about that tells us about .
and for . : :
Use these derivatives to find
and . Is getting steeper or less steep at ? At ? Explain your reasoning.
Consider this: Even though a slope of
is steeper than a slope of , if the slope changes from to , then the slope is decreasing.
The values in part (c) can be used to determine concavity. Where is
concave up? Where is concave down?
Positive values of the 2nd-derivative indicate that the function is concave up while negative values indicate that the function is concave down.