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**Spec** The requirement that the two gradients are parallel can also be expressed as

$\displaystyle \left|\begin{array}{cc}f'_x&f'_y\\g'_x&g'_y\end{ar ray}

\right|=0$

This will produce exactly the same result as using the Lagrange multipliers method (once you've eliminated the multiplier).

Both these methods are a bit impractical here though. Since we're dealing with an ellipse, we can express the main function in terms of one variable.

$\displaystyle f(x,y)=f(\varphi)=\left\{x=\cos\varphi,\ y=\frac{\sin \varphi}{\sqrt{2}} \right\}=\cos^2 \varphi+\frac{\sin^2 \varphi}{2}+\frac{2\cos \varphi\sin \varphi}{3}$

You will of course also need to find the inner points where $\displaystyle f'_x=f'_y=0$