Exercise 55

A function \(f\) is called homogeneous of degree n if it satisfies the equation \(f(tx, ty) = t^n f(x, y)\) for all \(t\), where \(n\) is a positive integer and \(f\) has continuous second-order partial derivatives. (a) Verify that \(f(x, y) = x^2y + 2xy^2 + 5y^3\) is homogeneous of degree 3. (b) Show that if \(f\) is homogeneous of degree \(n\), then \[ x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} = nf(x, y) \] [Hint: Use the Chain Rule to differentiate \(f(tx, ty)\) with respect to \(t\).]