Lagrange with e^-xy

[engineertobe]

Find the extreme values of f(x

y)=e−xy
on the region described by x2+25y2

16


I can get to -ye^(-xy)= λ2x, -xe^(-xy)= λ50y but then I do not know how to solve for λ so I become stuck

 

[galactus]

You have \(\displaystyle -ye^{-xy}=2x\lambda\Rightarrow \lambda=\frac{-ye^{-xy}}{2x}\)

\(\displaystyle -xe^{-xy}=50y\lambda\Rightarrow \lambda = \frac{-xe^{-xy}}{50y}\)

\(\displaystyle \frac{-ye^{-xy}}{2x}=\frac{-xe^{-xy}}{50y}\)

\(\displaystyle \frac{-y}{2x}=\frac{-x}{50y}\)

\(\displaystyle x=\pm 5y\)

Now, sub into the constraint and find x and y.