Mathematics · Calculus

Let f: ℝ2 → ℝ be a twice continuously differentiable non-zero function such that f(t x1, t x2) = t3 f(x1, x2) for all t > 0 and (x1, x2) ∈ ℝ2. Which of the following statements is necessarily true?

Verified Solution4 OptionsUpdated: October 2026👁️1 View

1∂f/∂x1(1,-1) - ∂f/∂x2(1,-1) = 3 f(1,-1)
23 ∂f/∂x1(1,1) + 3 ∂f/∂x2(1,1) = f(1,1)
3x12 ∂2f/∂x12(x1, x2) + x22 ∂2f/∂x22(x1, x2) + 2 x1 x2 ∂2f/∂x1∂x2(x1, x2) = 9 f(x1, x2)
4x12 ∂2f/∂x12(x1, x2) + x22 ∂2f/∂x22(x1, x2) + 2 x1 x2 ∂2f/∂x1∂x2(x1, x2) = 6 f(x1, x2)✓

Correct answer

x12 ∂2f/∂x12(x1, x2) + x22 ∂2f/∂x22(x1, x2) + 2 x1 x2 ∂2f/∂x1∂x2(x1, x2) = 6 f(x1, x2)

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