What are the critical points of # f(x,y)=sin(x)cos(y) +e^xtan(y)#?

1 Answer
Mar 20, 2018

Answer:

When #cos(x-y)+e^x(-tan^2(y)+tan(y)-1)=0#

Explanation:

We are given #f(x,y)=sin(x)cos(y) +e^xtan(y)#

Critical points occur when #(delf(x,y))/(delx)=0# and #(delf(x,y))/(dely)=0#

#(delf(x,y))/(delx)=cos(x)cos(y)+e^xtan(y)#

#(delf(x,y))/(dely)=-sin(x)sin(y)+e^xsec^2(y)#

#sin(y)sin(x)+cos(y)cos(x)+e^xtan(y)-e^xsec^2(y)=cos(x-y)+e^x(tan(y)-sec^2(y))=cos(x-y)+e^x(tan(y)-(1+tan^2(y)))=cos(x-y)+e^x(-tan^2(y)+tan(y)-1)#

There is no real way to find solutions, but critical points occur when #cos(x-y)+e^x(-tan^2(y)+tan(y)-1)=0#

A graph of solutions is here