Question

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

Answer 1

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