Is anyone able to explain why my solution to the following problem is not correct?
Maximize $f(x,y) = x^4y^8$ on the unit circle, with $x,y\ge0$
Clearly $f(x,y) \ge 0$ for any $x,y\in\mathbb R$, and thus on the unit circle a maximum will occur when neither $x$ nor $y$ are $0$. So, consider the case where $x\ne0, y\ne 0$. Then, we set $$f_x(x,y) = 4x^3y^8 = 0,$$ $$f_y(x,y) = 8x^4y^7 = 0,$$ which implies that $$4x^3y^8 = 8x^4y^7 \implies y=2x,$$ since $x\ne0,y\ne0$, we may divide through by $x$ and $y$. Since we are on the unit circle, we substitute this value into $$x^2+y^2 = 1$$ $$x^2 + (2x)^2 = 1$$ $$x^2 + 4x^2 = 1$$ $$5x^2 = 1$$ $$x = \sqrt{1\over5}$$ $$\implies y = 2\sqrt{1\over5}.$$
According to Wolfram|Alpha, the point on the unit circle is supposed to be $$\left({1\over\sqrt3},{\sqrt{2\over3}}\right).$$ Logically my steps seem alright, but I just can't quite figure out what happened. One of the students I TA for brought me this question and it just bugged me that I could never resolve it.