Let $x$ and $y$ be real numbers such that $0 \le x \le 1$ and $0 \le y \le 1.$  Find the maximum value of
\[x^2 y - xy^2.\]
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Let $x$ and $y$ be real numbers such that $0 \le x \le 1$ and $0 \le y \le 1.$  Find the maximum value of
\[x^2 y - xy^2.\]
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
\frac{1}{4}
Let $x$ and $y$ be real numbers such that $0 \le x \le 1$ and $0 \le y \le 1.$  Find the maximum value of
\[x^2 y - xy^2.\]
train