You know that the first derivative,
What does the second derivative, f ″, tell you about f ′? What does
tell you about ? tells us the same thing about that tells us about .
Write equations for
and if . : :
getting steeper or less steep at ? At ? Use your derivatives from part (b) to 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 of
and can be used to determine concavity at and . 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.