Answer
$f(x,y,z)=x^2+\dfrac{3}{2}y^2+2z^2$
Work Step by Step
Since, we have $\nabla f =F$
and $f(x,y,z)=x^2+g(y,z)$ ...(1)
Here, $g_y(y,z)=3y \implies g(y,z)=\dfrac{3}{2}y^2+h(z) $ and $h(z)=2z^2+C$
From equation (1), we have
$f(x,y,z)=x^2+\dfrac{3}{2}y^2+h(z)=x^2+\dfrac{3}{2}y^2+2z^2$
or, $f(x,y,z)=x^2+\dfrac{3}{2}y^2+2z^2$