If there is such a scalar function f, then
[tex]\dfrac{\partial f}{\partial x}=4y^2[/tex]
[tex]\dfrac{\partial f}{\partial y}=8xy+4e^{4z}[/tex]
[tex]\dfrac{\partial f}{\partial z}=16ye^{4z}[/tex]
Integrate both sides of the first equation with respect to x :
[tex]f(x,y,z)=4xy^2+g(y,z)[/tex]
Differentiate both sides with respect to y :
[tex]\dfrac{\partial f}{\partial y}=8xy+4e^{4z}=8xy+\dfrac{\partial g}{\partial y}[/tex]
[tex]\implies\dfrac{\partial g}{\partial y}=4e^{4z}[/tex]
Integrate both sides with respect to y :
[tex]g(y,z)=4ye^{4z}+h(z)[/tex]
Plug this into the equation above with f , then differentiate both sides with respect to z :
[tex]f(x,y,z)=4xy^2+4ye^{4z}+h(z)[/tex]
[tex]\dfrac{\partial f}{\partial z}=16ye^{4z}=16ye^{4z}+\dfrac{\mathrm dh}{\mathrm dz}[/tex]
[tex]\implies\dfrac{\mathrm dh}{\mathrm dz}=0[/tex]
Integrate both sides with respect to z :
[tex]h(z)=C[/tex]
So we end up with
[tex]\boxed{f(x,y,z)=4xy^2+4ye^{4z}+C}[/tex]
