6.5. Differentiable Functions
Corollary 6.5.13: Finding Local Extrema
Suppose
f is differentiable on
(a, b). Then:
 If f'(c) = 0 and f'(x) > 0 on (a, x) and
f'(x) < 0 on (x, b), then f(c) is a local maximum.
 If f'(c) = 0 and f'(x) < 0 on (a, x) and
f'(x) > 0 on (x, b), then f(c) is a local minimum.
Context
Proof:
This corollary becomes obvious when we interpret what it means
for the function to have a positive or negative derivative, as
in these tables:
Loc. Max 
interval  (a, c)  (c , b) 
sign of f'(x)  +   
dir. of f(x)  up  down 

Loc. Min 
interval  (a, c)  (c , b) 
sign of f'(x)    + 
dir. of f(x)  down  up 

No Extremum 
interval  (a, c)  (c , b) 
sign of f'(x)  +  + 
dir. of f(x)  up  up 

No Extremum 
interval  (a, c)  (c , b) 
sign of f'(x)     
dir. of f(x)  down  down 

Of course, these tables are no proof  which is once again left as an exercise.