Consider w = F(x,y,z) = x2 + yz3Find the equation of the tangent when x= 3, y = 2, z = 1. At (3,2,1), in what direction is w increasing most rapidly?

The given surface is Fx,y,z=x2+yz3. (1) The given point is x=3,y=2, z=1. We know that the equation of tangent plane to the surface Fx,y,z at x=a, y=b, z=c can be calculated as: x-a∂F∂xa,b,c+y-b∂F∂ya,b,c+z-c∂F∂za,b,c=0. Since the given point is x=3,y=2, z=1, therefore a=3,b=2, c=1. Since Fx,y,z=x2+yz3, therefore: ∂F∂x=2x∂F∂y=z3∂F∂z=3yz2Therefore at a=3,b=2, c=1, we have: ∂F∂x3,2,1=23=6∂F∂y3,2,1=13=1∂F∂z3,2,1=3212=6 Therefore the tangent plane at a=3,b=2, c=1 can be calculated as: x-3∂F∂x3,2,1+y-2∂F∂y3,2,1+z-1∂F∂z3,2,1=0x-36+y-21+z-16=06x-18+y-2+6z-6=06x+y+6z=26 Hence the required tangent plane is 6x+y+6z=26.(2) We know that a function w=Fx,y,z increases most rapidly in a direction its gradient. Since the given point is 3,2,1, therefore from the previous part, we have; ∂F∂x3,2,1=6∂F∂y3,2,1=1∂F∂z3,2,1=6 Therefore the direction of gradient at 3,2,1 is ∇F=6i^+1j^+6k^. Therefore the direction of increasing most rapidly is: d→=6i^+1j^+6k^62+12+62=1736i^+1j^+6k^ Hence the required direction, where W=Fx,y,z increases most rapidly is 1736i^+1j^+6k^.