f(x) Compared to its First Derivative
f(x) Compared to its Second Derivative
f(x) = x6 - 3x5 + 6x3 - 3x2 - 3x + 2
f(x) = x6 - 3x5 + 6x3 - 3x2 - 3x + 2
3 y
local
maximum
3 y
2
2
inflection
point
1
-1
1
local
minimum
-1
inflection
points (4)
1
2
x
-1
local
minimum
concave
up
1
-1
concave
down
2
x
concave concave
down
up
concave
up
6 f‛(x)
4
10 f‛‛(x)
2
-1
5
1
2
x
-1
-2
-5
-4
-10
-6
-15
graph feature
(L to R)
slope > 0
falling
(L to R)
slope < 0
maximum
slope = 0
minimum
slope = 0
inflection pt.
concave up
concave down
f‛(x)
f(x)
rising
x
4 zeros
in the second
derivative graph
f‛‛(x)
+
=0
+ on L
- on R
=0
- on L
+ on R
Curvature changes:
↔
2
f‛‛(x) = 30x4 - 60x3 + 36x - 6
f‛(x) = 6x5 - 15x4 + 18x2 - 6x - 3
extrema
1
+
=0
+
+
-
at xmax
at xmin
potential
inflection point
+
-
Notes
derivative may not
exist at a max or
min, e.g.
or