Discussion
For the homogeneous function \( f = x^{3} + y^{3} \), the value of \( x f_{x} + y f_{y} \) is
Answer: A
The function is homogeneous of degree three, so Euler's theorem applies with \( n = 3 \). Checking it directly,
\[ x f_{x} + y f_{y} \;=\; x\left(3x^{2}\right) + y\left(3y^{2}\right) \;=\; 3\left(x^{3} + y^{3}\right) \;=\; 3f. \]
The degree, not the number of variables, sets the multiplier, which is why a quadratic form would return \( 2f \)
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