Modul
e
4
: Local / Global Maximum / Minimum and Curve Sketching
Lecture
11 : Convex / Concave Functions [Section 11.2]
11.2.8
Theorem
(Second derivative test for convexity / concavity):
Let
be such that
exists in
. Then the following hold:
(i)
concave upward if and only if
for all
.
(ii)
concave downward if and only if
for all
.
(iii)
If
for all
, then
is strictly concave upward.
(iv)
If
for all
, then
is strictly concave downward.
Proof:
Follows from theorem 11.2.6 and 10.1.1
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be such that 
exists in 
. Then the following hold: 
concave upward if and only if 
for all 
. 
concave downward if and only if 
for all 
. 
for all 
, then 
is strictly concave upward. 
for all 
, then 
is strictly concave downward.
