Maximum Possible Value

Algebra Level 3

x , y x, y and z z are non-negative real numbers such that x + y + z = 1 x+y+z=1 .

What is the maximum possible value of x + y 2 + z 3 x+y^2+z^3 ?

3 2 0 1 4

Hana Wehbi by Hana Wehbi , 51, USA

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1 solution

Hana Wehbi
Nov 5, 2016

Since 0 ≤ y , z ≤ 1 0 \le y, z\le 1 , we have y 2 ≤ y a n d z 3 ≤ z y^2 \le y \ and \ z^3 \le z . Therefore, x + y 2 + z 3 ≤ x + y + z = 1 x+y^2+z^3 \le x+y+z=1 .

We can get x + y 2 + z 3 = 1 x+y^2+z^3=1 by setting ( x , y , z ) = ( 1 , 0 , 0 ) (x,y,z) = (1,0,0) .

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