Let $x$ and $y$ be nonnegative real numbers such that $x + y = 1.$  Find the maximum value of
\[x^4 y + xy^4.\]
Question:
Let $x$ and $y$ be nonnegative real numbers such that $x + y = 1.$  Find the maximum value of
\[x^4 y + xy^4.\]
Answer:
Let's think step by step.

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\frac{1}{12}
Let $x$ and $y$ be nonnegative real numbers such that $x + y = 1.$  Find the maximum value of
\[x^4 y + xy^4.\]
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