Let $w,$ $x,$ $y,$ and $z,$ be positive real numbers.  Find the maximum value of
\[\frac{wx + xy + yz}{w^2 + x^2 + y^2 + z^2}.\]
Question:
Let $w,$ $x,$ $y,$ and $z,$ be positive real numbers.  Find the maximum value of
\[\frac{wx + xy + yz}{w^2 + x^2 + y^2 + z^2}.\]
Answer:
Let's think step by step.

user
\frac{1 + \sqrt{5}}{4}
Let $w,$ $x,$ $y,$ and $z,$ be positive real numbers.  Find the maximum value of
\[\frac{wx + xy + yz}{w^2 + x^2 + y^2 + z^2}.\]
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