Euler relations for a homogeneous function of degree 3

Let $f:\mathbb R^2\to\mathbb R$ be a twice continuously differentiable non-zero function such that

$$f(tx_1,tx_2)=t^3f(x_1,x_2)$$

for all $t>0$ and $(x_1,x_2)\in\mathbb R^2$. Which of the following statements are necessarily true?