#f(x)=e^x[6-e^x]#
#D(f)=RR#
Doesn't have a vertical asymptote
#f(-x)=e^-x[6-e^-x]!=f(x)!=-f(x)#
This function isn't odd and neither even
isn't periodic function (han no sinx, cosx...)
Inceptions:
If #x=0 => y=e^0[6-e^0]=5# .......[0,5]
If #y=0 => 0=e^x[6-e^x]#
#e^x>0# so we have to decide for #(6-e^x)#
#6-e^x=0 =>6=e^x#
#ln6=x=1.792#.................................[ln6,0]
#f^'(x)=[6e^x-e^(2x)]^'#
#f^'(x)=6e^x-2e^(2x)=2e^x(3-e^x)#
#2e^x>0# so we decide for #(3-e^x)#
#3-e^x=0#
#3=e^x#
#ln 3=x=1.098...#
#x in (-oo,ln3)hArr f^'(x)>0 => f uarr#
#x=ln3 => "maximum"#
#x in (ln3,oo)hArr f^'(x)<0 => f darr#
#f^''=[6e^x-2e^(2x)]^'#
#f^''=6e^x-4e^(2x)#
#f^''=2e^x(3-2e^x)#
again #2e^x>0# so we decide for #(3-2e^x)#
#3-2e^x=0#
#3/2=e^x#
#ln (3/2)=x=0.4054...#
#x in (-oo,ln(3/2))hArr f^''(x)>0 => "f(x) is convex" uu#
#x in (ln(3/2),oo)hArr f^''(x)<0 => "f(x) is concave" nn#
Behaving in the #+oo#
#Lim_(xrarroo)e^x[6-e^x]=oo*(-oo)=-oo#
Behaving in the #-oo#
#Lim_(xrarr-oo)e^x[6-e^x]=0*(6-0)=0#
The range: Since maximum is in the point #x=ln3# we can calculate that:
#f(ln3)=e^ln3[6-e^(ln3)]=3[6-3]=9#
#H(f)=(-oo,9>#
