The slope of the graph is decreasing for
x < 2
, so in this region
f''(x) < 0
. Similarly, the
slope of the graph is increasing for
x > 2
, so
f''(x) > 0
here. Thus at
x = 2
, we
must have
f''(x) = 0
(by the intermediate value theorem).
The second derivative being zero means that the slope of the graph of the function is
constant for all
x
. The only graphs for which this is true are the lines, so
f (x) = mx + b
for some constants
m
and
b
.