<s>[INST] maximum of $x^2-3xy-2y^2$ subjects to $x^2+xy+y^2=1$ without Lagrange multiplier What is the maximum of $x^2-3xy-2y^2$ subjects to $x^2+xy+y^2=1$? I wonder there is a precalculus method, without using the Lagrange multiplier.
 [/INST] $$x^2+xy+y^2=1\implies y_\pm=-\frac{1}{2} \left(x \pm\sqrt{4-3 x^2}\right)$$
Using $y_-$ gives
$$f=x^2-3xy_m-2y_m^2=\frac{7 x^2}{2}+\frac{1}{2} \sqrt{4-3 x^2} x-2$$ Compute the derivative, set it equal to $0$ and solve for $x$. You should get
$$x_1=\sqrt{\frac{1}{39} \left(26+7 \sqrt{13}\right)} \qquad x_2=-\sqrt{\frac{1}{39} \left(26-7 \sqrt{13}\right)}$$ Compute the coreesponding $y_1$ and $y_2$.
Do the same with $y_+$.
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