A.
#"Total surface area"="Surface area of cylinder"+"Surface area of two hemispheres"#
#color(white)("Total surface area")="Surface area of cylinder"+"Surface area of a sphere"#
#color(white)("Total surface area")=y(2pix)+4pix^2#
#color(white)("Total surface area")=2pixy+4pix^2#
#A=2pixy+4pix^2#
B.
#pi/6=(4pix^3)/3+ypix^2#
#1/6=(4x^3)/3+yx^2#
#yx^2=1/6-(4x^3)/3#
#y=(1/6-(4x^3)/3)/x^2=1/(6x^2)-(4x)/3#
C.
#A=2pix(1/(6x^2)-(4x)/3)+4pix^2#
#color(white)(A)=(2pix)/(6x^2)-(2pix(4x))/3+4pix^2#
#color(white)(A)=pi/(3x)-(8pix^2)/3+4pix^2#
#color(white)(A)=pi/(3x)+4pix^2-(8pix^2)/3#
#color(white)(A)=pi/(3x)+(12pix^2-8pix^2)/3#
#color(white)(A)=pi/(3x)+(4pix^2)/3#
D.
#A=pi/(3x)+(4pix^2)/3#
#(dA)/(dx)=d/dx[pi/(3x)+(4pix^2)/3]#
#color(white)((dA)/(dx))=d/dx[pi/(3x)]+d/dx[(4pix^2)/3]#
#color(white)((dA)/(dx))=pi/3d/dx[x^(-1)]+(8pix)/3#
#color(white)((dA)/(dx))=pi/3(-x^(-2))+(8pix)/3#
#color(white)((dA)/(dx))=-pi/(3x^2)+(8pix)/3#
#-pi/(3x^2)+(8pix)/3=0#
#(8pix)/3=pi/(3x^2)#
#(8x)/3=1/(3x^2)#
#3x^2(8x)=3#
#8x^3=1#
#x^3=1/8#
#color(white)(x)=1/root(3)(8)#
#color(white)(A)x=1/2#
#A=pi/(3(1/2))+(4pi(1/2)^2)/3#
#color(white)(A)=pi/(3/2)+pi/3#
#color(white)(A)=(2pi)/3+pi/3#
#color(white)(A)=picm^2#