Chapter 4 Local Max & Mins & Inflection Points
For xxxxf1232)( 23 −−= , determine:
a) Critical values.
b) Critical points.
c) Intervals on the x-axis where f(x) is increasing and decreasing.
d) Intervals on the x-axis where f(x) is concave up and concave down.
e) Determine point(s) of inflection, if any.
f) The relative maximum and relative minimum values of f(x).
Solutions:
a) Critical values.
The critical values are found by setting f ' (x) = 0 and solving for x.
Here f ' (x) = 6x2 - 6x -12 then we need to solve f ' (x) = 6x2 - 6x -12 = 0.
6(x2 -x -2) = 6 (x-2)(x+1) = 0 ⇒ x = 2 & x = -1 are the critical values.
b) Critical Points.
Critical points are points on the graph of the function, f(x). So we need to substitute -1 & 2
into f(x) to get the coordinates.
(2, -20) & (-1, 7) are the critical points.
C) Intervals where f(x) is increasing and decreasing.
Recall, f ' (x) = 6x2 - 6x -12 = 6(x-2)(x+1), we will use the first derivative test here.
0 0
+ - + f '
-1 2
