Multivariable extrema on restricted domain

Question

I want to find $max/min$ of the function $f(x,y) = e^{3xy}$ in the first quadrant limited by the lines $y=2x$, $x=2y$ and the curves $xy=2$ and $xy=6$

Since $f(x,y)$ only has a saddle point, I guess I have to check the boundaries, but how do I go about doing that in this situation?

thank you

Answer 1

Find the intersection point between the line which define the boundaries and then consider

• $y=2x\implies f(x,2x) = e^{6x^2}$ for $x\in[1,\sqrt 3]$
• $x=2y\implies f(2y,y) = e^{6y^2}$ for $y\in[1,\sqrt 3]$
• $xy=2\implies f(x,2/x) = e^{6}$
• $xy=6\implies f(x,6/x) = e^{18}$