As you know, the volume of a cylinder is 

V = , 

where pi = 3.14, r is the radius and h is the height.

In your case the volume is fixed:

 = 64.         (1)


The surface area of a cylinder is 

S = ,    (2)

and they ask you to find minimum of (2) under the restriction (1).


I can rewrite (2) in the form

S(r) =  +  =  + .   (1)


The plot below shows the function S(r) =  + , and you can clearly see that it has the minimum.





Plot y = 


To find the minimum, use Calculus: differentiate the function to get

S'(r) =  +  = 

and equate it to zero.


S'(r) = 0   lead you to the equation   = ,   which gives 

r =  =  = 1.72 (approximately).


Answer.  r = 1.72 units, h =  units give the minimum of the surface area.