Answer
$$\frac{\partial^2 f}{\partial x^2}=6 y,\quad
\frac{\partial^2 f}{\partial y^2}= -72xy^2.$$
Work Step by Step
Since $ f(x,y)=3x^2y-6xy^4$, then we have
$$\frac{\partial f}{\partial x}=6xy-6y^4,\quad
\frac{\partial f}{\partial y}=3x^2 -24xy^3.$$
Hence, we get
$$\frac{\partial^2 f}{\partial x^2}=6 y,\quad
\frac{\partial^2 f}{\partial y^2}= -72xy^2.$$
