Find the maximum of $f(x, y) = \sum_{k = 1}^{n} x_k y_k$ subject to $\sum_{k=1}^{n} x_k^2 = 1$ and $\sum_{k=1}^{n} y_k^2 = 4$

I just applied Cauchy Schwarz inequality to find

$$-4 \leq |f(x, y)| \leq 4. $$

So I believe the maximum is $4$ and the minimum is $-4$. The maximum is attained when $x = (1, 1)$ and $y = (2, 2)$. The minimum can attained when $x = (-1 , -1)$ and $y = (2, 2)$.

Is this enough to prove that my bounds are optimal? If not, how can I do so?

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    $\begingroup$ Your $y$ does not satisfy $\sum y_k^2 = 4$. $\endgroup$ – kccu May 20 at 19:05
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    $\begingroup$ Why have you deleted your previous question I really tried to help you. $\endgroup$ – callculus May 20 at 19:21
  • $\begingroup$ @callculus I undeleted it. My original post is the correct answer, so I didn't think it was necessary to leave it up. $\endgroup$ – gallileo22 May 20 at 19:29
  • $\begingroup$ @gallileo22 In my view the right answer depends on some assumptions which I´ve asked for. The question itself is interesting. $\endgroup$ – callculus May 20 at 19:37

Cauchy-Schwarz actually gives you a bound of 2, not 4. In this case $x_i = \frac{1}{\sqrt{n}}$ and $y_i = \frac{2}{\sqrt{n}}$ suffice for the maximum, and negating all the $x_i$ produces the minimum.

  • $\begingroup$ So how do we show it's optimal? $\endgroup$ – gallileo22 May 20 at 19:39

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