What is the largest

Algebra Level 3

If x x and y y are real numbers satisfying x 2 + y 2 = x + y x^2+y^2 =x+y , find the maximum value of x + y x+y .


The answer is 2.

Choi Chakfung by choi chakfung , 28, Hong Kong

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2 solutions

Using Cauchy-Schwarz inequality, we have:

( x + y ) 2 ≤ ( 1 + 1 ) ( x 2 + y 2 ) ≤ 2 ( x + y ) ⇒ x + y ≤ 2 \begin{aligned} (x+y)^2 & \le (1+1)(x^2+y^2) \\ & \le 2(x+y) \\ \Rightarrow x+y & \le \boxed{2} \end{aligned}

Equality occurs when x = y = 1 x=y=1 .

1 Helpful 0 Interesting 1 Brilliant 0 Confused

x + y x+y will be maximum when both are positive.

Therefore applying R M S ≥ A M RMS \geq AM

We have ( x 2 + y 2 ) / 2 ≥ ( x + y ) / 2 \sqrt{( x^{2} + y^{2} )/2} \geq (x+y)/2 .

Hence 2 ≥ x + y 2 \geq x+y

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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