Question

Please help me with the following problem. I am asked to show that f does not have an absolute maximum.

f(x,y)=3x(e^y)-x^3-e^(3y)

Answer 1

I found the relative max at the critical point (1,0) but I don't know how to prove that f does not have an absolute extrema.

Answer 2

when you solve for that critical point , do you notice that the denominator has (x -1)? that means reject the value of x =1. You got the critical point (1,0) and reject x =1---> no critical point, right?

Answer 3

May be I am wrong, but I got \[\huge y'=\frac{x^2-e^y}{e^y(x-1)}\]and solve for critical point , I got exactly what you got . We should consider the denominator of y', right?