Question #46a96

1 Answer
Dec 9, 2017

Explanation below
graph{6e^(x)-e^(2x) [-14.19, 14.29, -5.05, 9.78]}

Explanation:

#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>#