<s>[INST] If $x$ and $y$ are real and $x^2 + y^2 = 1$, compute the maximum value of $(x+y)^2.$ If $x$ and $y$ are real and $x^2 + y^2 = 1$, compute the maximum value of $(x+y)^2.$

Should I manipulate the equation?  I'm stuck on this problem.  Answers are greatly appreciated.
 [/INST] Use the Cauchy-Schwarz inequality:
$$(x\times 1+y\times 1 )^2 \leq (x^2+y^2)(1^2+1^2), x^2+y^2=1$$
$$ (x+y)^2 \leq (1)(2)$$
$$-\sqrt{2} \leq x+y \leq \sqrt{2} $$
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