Consider the function . It is easy to check (using definition or proportion 11.2.3) that it is concave up everywhere.
If we look at the slop of tangent, i.e., , we observe that it increases as we move from left to right. Analytically, is increasing for all .
Similarly, for the function which is concave down, the slop of the tangent, decreases as we move from left to right.
This motivates our next theorem.
11.2.6
Theorem (First derivative test for convexity / concavity):
Let be such that exists. Then the following holds:
(i) is concave upward if and only if is increasing.
(ii) is concave downward if and only if is decreasing.
(iii) If is strictly increasing, then is strictly concave upward.
(iv) If is strictly decreasing, then is strictly concave upward.
. It is easy to check (using definition or proportion 11.2.3) that it is concave up everywhere. 
, we observe that it increases as we move from left to right. Analytically, 
is increasing for all 
. 
which is concave down, the slop of the tangent, 
decreases as we move from left to right. 
be such that 
exists. Then the following holds: 
is concave upward if and only if 
is increasing. 
is concave downward if and only if 
is decreasing. 
is strictly increasing, then 
is strictly concave upward.
is strictly decreasing, then 
is strictly concave upward. 
