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:

     = 1000 cubic centimeters.                   (1)


The surface area of a cylinder is 

    S =  + ,                               (2)

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


You can rewrite the formula (2) in the form

    S(r) =  + .                         (3)


In formula (3), replace    by  1000, based on (1). You will get

    S(r) =  +  =  + .


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   leads you to equation   = ,   which gives 

r =  =  = 5.42 cm (approximately).


Answer.  r = 5.42 cm, h =  = 10.84 cm  give the minimum of the surface area.