What is the graphic for #f(x,y)=x y^(x/y)=C_0# ?

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Answer 1 · 2016-11-04T10:28:36

See below.

Making #x = lambda y# and substituting in

#f(x,y)=x y^(x/y)=c_0# we have

#f(lambda y,y)=lambda y y^lambda=lambda y^(lambda+1)=c_0#

so

#y(lambda)=(c_0/lambda)^(1/(lambda+1))# and also

#x(lambda)=lambda (c_0/lambda)^(1/(lambda+1))# so we can generate #x,y# pairs obeying the condition.

#f(x,y)=x y^(x/y)=c_0#

Attached a plot of

#x(lambda),y(lambda)# for #lambda in [0.1,3]#

enter image source here