What are the critical points of # f(x) = e^(x^(1/3))*sqrt(x+8)#?

1 Answer
Apr 8, 2018

Answer:

#f(x)^/=##(3)##(1)##+##(2)##(4)#

please see below !

Explanation:

#f(x)=e^(x^(1/3))*sqrt(x+8)#

#y=sqrt(x+8)# #(1)#

#z=e^(x^(1/3))# #(2)#

#f(x)=z*y#

#f(x)^/=z^/*y+z*y^/#

#z=e^(x^(1/3))#

#x^(1/3)=n#

#z=e^n#

#x^(1/3)=n#

#(dn)/dx=1/3x^(-2/3)#

#z=e^n#

#dz/(dn)=e^n#

#(dz)/(dn) * (dn)/dx# = #(dz)/dx#

#(dz)/dx=# #1/3x^(-2/3)*e^(x^(1/3))# #(3)#

#y=sqrt(x+8)#

#u=x+8#

#(du)/(dx)=1#

#y=sqrt(u)#

#dy/(du)=1/2u^(-1/2)#

#dy/(du)*##(du)/(dx)##=dy/dx#

#dy/dx=1/2(x+8)^(-1/2)# #(4)#

#f(x)^/=z^/*y+z*y^/#

#f(x)^/=##(3)##(1)##+##(2)##(4)#