A function f is increasing (or strictly increasing) if f (a) < f (b)
whenever a < b.
When you move from left to right (from a to b since a < b) along
the graph the value of the function increases (f (a) < f (b)).
I
f is increasing on the interval I.
A function f is decreasing (or strictly increasing) if f (a) > f (b)
whenever a < b.
When you move from left to right (from a to b since a < b) along
the graph the value of the function decreases (f (a) > f (b)).
I1
I2
f is decreasing on the intervals I1 and I2 .
A point a in the domain of a function f is a relative maximum
point of f if the function the function changes from increasing
to decreasing at a.
Il
a
Ir
f is increasing on the interval Il (to the left of a)
and decreasing on the interval Ir (to the right of a).
A point a in the domain of a function f is a relative minimum
point of f if the function the function changes from decreasing
to increasing at a.
Il
a
Ir
f is decreasing on the interval Il (to the left of a)
and increasing on the interval Ir (to the right of a).
The relative maximum points and the relative minimum points
of f together make up the relative extreme points or relative
extrema.
decreasing
increasing
relative extreme point
decreasing
relative extreme point
There are two relative extrema. One is a relative minimum
point and the other is a relative maximum point.
A function f is concave up at a point a if there is an open
interval around a in which the graph of the function lies above
the tangent line at a.
(
a
)
There is an open interval around a in which the graph of the
function lies above the tangent line at a.
A function f is concave down at a point a if there is an open
interval around a in which the graph of the function lies below
the tangent line at a.
(
a
)
There is an open interval around a in which the graph of the
function lies below the tangent line at a.
We will use the following fact: A function f is concave up
at a point a if there is an open interval around a in which the
slope of the tangent line increases as x increases.
(
a
)
The slope of the tangent line to the graph increases as x
increases within an interval around a.
Also, a function f is concave down at a point a if there is
an open interval around a in which the slope of the tangent line
decreases as x increases.
(
a
)
The slope of the tangent line to the graph decreases as x
increases within an interval around a.
The First Derivative Rule If f 0 (a) > 0, then f (x) is increasing
at x = a. If f 0 (a) < 0, then f (x) is decreasing at x = a
The Second Derivative Rule If f 00 (a) > 0, then f (x) is concave up at x = a. If f 00 (a) < 0, then f (x) is concave down at
x=a
We can use these two rules to determine the shape of the graph
of a function f around a point a based on the values of the first
and second derivative of f at x = a.