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.

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.

                                                

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


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