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.
[Math] If $x$ and $y$ are real and $x^2 + y^2 = 1$, compute the maximum value of $(x+y)^2.$
algebra-precalculusoptimization
Related Solutions
This question is an old problem posed on Brilliant.
You can view it here, along with the solution by the problem creator Zi Song.
Let $\vec a=(2,-5), \vec b=(x,y)$.
$2x-5y=\vec a\cdot \vec b=\sqrt{2^2+(-5)^2}\cdot\sqrt{x^2+y^2}\cos\theta\le\sqrt{29}$ where $\theta$ is the angle between the two vectors.
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Best Answer
Trig-free solution: If $x^2+y^2=1$ then $$ (x+y)^2 + (x-y)^2 = 2(x^2+y^2)=2 $$ so the largest value of $(x+y)^2$ is $2$, attained when $x=y$.