For a given volume of a riser, if the solidification time of the molten metal in riser needs to be quadrupled, the surface area of the riser should be made

The system of equations, given below, has $$$x+2y+4z=2$$$ $$$4x+3y+z=5$$$ $$$3x+2y+3z=1$$$

Directional derivative of $$\phi = 2xz - {y^2}$$ at the point $$(1, 3, 2)$$ becomes maximum in the direction of

If $$\,\phi = 2{x^3}{y^2}{z^4}$$ then $${\nabla ^2}\phi $$ is