Let be any unit vector.
then
If both and exist and has a local maximum or a local
minimum at , then
Suppose has a local maximum at . Then, we can find some such that Hence, if exists, then Similarly, . Hence, The case of a local minimum is similar. This proves (i). To prove (ii), we note that if exists and if , then Considering , we similarly obtain .
be any unit vector. 
has a local maximum/local minimum at 
and 
exists, 
and 
exist and 
has a local maximum or a local 
, then 
has a local maximum at 
. Then, we can find some 
such
that 
exists, then 
. Hence, 
exists and if 
, then 
, we similarly obtain 
. 
