Concavity and the relationship between f,f′ and f″
Concavity
Definitions:
- If the graph of f lies above all of its tangent
lines on an open interval, the we say it is concave
up on that interval.
- If the graph of f lies below all of its tangent
lines on an open interval, then we say it is concave
down on that interval.
- A point, P, on a continuous curve f(x) is an inflection
point if f changes concavity there.
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When a curve is concave up, it is sort
of bowl-shaped, and you can think it might hold water, with the rim
pointing up. Notice that a function that is concave up may be increasing or
decreasing:

Similarly, when a curve is concave down,
it is sort of upside-down-bowl-like, and water would run off of it,
with the rim pointing down. Notice
that a function that is concave down may be
increasing or decreasing:
f, f' and f''
When the slopes of tangent lines are increasing, i.e. when f'
is increasing, the function is concave up, as you can see below in
the first two graphs. Since (f')'=f'', when f' is
increasing, f'' is positive. Similarly, when the slopes of
tangent lines are decreasing, i.e. when f' is decreasing, the
function is concave down, as you can see in the second two graphs
below. Since (f')'=f'', when f' is decreasing, f'' is
negative.
DO: Spend some time
looking at the slopes and concavity below, and work to totally
understand the relationships between f,f' and f'' indicated
below the graphs.

\Large{f''>0\quad\Longleftrightarrow\quad f'\uparrow\quad
\Longleftrightarrow\quad f \cup}

\Large{
f''<0\quad\Longleftrightarrow\quad f'\downarrow\quad
\Longleftrightarrow\quad f \cap}
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