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

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Answer 1 · 2018-03-20T14:12:49

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

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

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