(i) Since exists and has a point of inflection at exist in for some such that, say, is strictly increasing in and is strictly decreasing in . Then, using definition
Hence . The case when is strictly decreasing in and is strictly increasing in can be analyzed similarly. This proves (i).
To prove (ii) let . The proof of the case where is similar. Since
there exists such that
Hence
Thus, by theorem 11.2.10, has a point of inflection at Back
exists and 
has a point of inflection at 
exist in 
for some 
such that, say, 
is strictly increasing in 
and is strictly decreasing in 
. Then, using definition 
. The case when 
is strictly decreasing in 
and 
is strictly increasing in 
can be analyzed similarly. This proves (i). 
. The proof of the case where 
is similar. Since 
such that 
has a point of inflection at 
