Use Lagrange multipliers to find the maximum and minimum values of the function :$f(x,y)=e^{xy}$ constraint $x^3+y^3=16$
This is my problem in my workbook. When I solve, I'm just have one solution, so I cannot find other. Here is my solution:
$f(x,y) = e^{xy}$ $g(x,y) = x^3+y^3-16$ $t(x,y) = f(x,y) + \lambda*g(x,y)$ So we will have three equations by Lagrange Multiplier:
$(1) y*e^{xy} + 3*\lambda*x^2 = 0$ $(2) x*e^{xy} + 3*\lambda*y^2 = 0$ $(3)x^3+y^3 = 16$
If $x=0$ or $y=0$ $==>$ $y=0$ or $x=0$ --> false if $t=0$ $==>$ $x=y=0$ ---> false
So, $x$,$y$ and $t$ cannot equal to 0.
So, we have from (1) (2) and (3): $\frac{e^{xy}}{-3*\lambda} = \frac{x^2}{y}$ $\frac{e^{xy}}{-3*\lambda} = \frac{y^2}{x}$ $==> x= y $ ==>$ x = y = 2 $
That is my solution. I just have one no, so I cannot find both min and max. Maybe something wrong with my solution. Please helps me.
Thanks :)