Let ϕ(x,y,z)=x2yz+4xz2𝜙(𝑥,𝑦,𝑧)=𝑥2𝑦𝑧+4𝑥𝑧2. The greatest rate of change of this function at the point (1,-2,1) is given bya.39−−√39b.None of thesec.31−−√31d.37−−√

The greatest rate of change of a function at a given point is given by the magnitude of the gradient of the function at that point.

The gradient of a function is a vector that points in the direction of the greatest rate of change of the function. It is given by the vector of the partial derivatives of the function with respect to each variable.

So, first we need to find the partial derivatives of the function ϕ(x,y,z)=x^2yz+4xz^2 with respect to x, y, and z.

The partial derivative of ϕ with respect to x is given by: ϕ_x = 2xyz + 4z^2

The partial derivative of ϕ with respect to y is given by: ϕ_y = x^2z

The partial derivative of ϕ with respect to z is given by: ϕ_z = x^2y + 8xz

Now we evaluate these at the point (1,-2,1) to get the gradient of ϕ at that point:

ϕ_x(1,-2,1) = 2(1)(-2)(1) + 4(1)^2 = -4 + 4 = 0 ϕ_y(1,-2,1) = (1)^2(1) = 1 ϕ_z(1,-2,1) = (1)^2(-2) + 8(1)(1) = -2 + 8 = 6

So the gradient of ϕ at (1,-2,1) is the vector (0, 1, 6).

The magnitude of this vector is given by the square root of the sum of the squares of its components, which is sqrt(0^2 + 1^2 + 6^2) = sqrt(1 + 36) = sqrt(37).

So, the greatest rate of change of the function ϕ(x,y,z)=x^2yz+4xz^2 at the point (1,-2,1) is sqrt(37). Therefore, the correct answer is d. sqrt(37).