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Algebra Difficulty 4.9 AIME Find the answer

Suppose that x,yx, y, and zz are non-negative real numbers such that x+y+z=1x+y+z=1. What is the maximum possible value of x+y2+z3x+y^{2}+z^{3} ?

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Since 0≤y,z≤10 \leq y, z \leq 1, we have y2≤yy^{2} \leq y and z3≤zz^{3} \leq z. Therefore x+y2+z3≤x+y+z=1x+y^{2}+z^{3} \leq x+y+z=1. We can get x+y2+z3=1x+y^{2}+z^{3}=1 by setting (x,y,z)=(1,0,0)(x, y, z)=(1,0,0).

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