In what direction from the point (1,2,1) is the directional derivative of  `\varphi  = xy^2z + x^2y` a maximum? What is the magnitude of this maximum?                                                    

Solution:

To find the direction in which the directional derivative of the function `\varphi = xy^2z + x^2y` is a maximum at the point (1, 2, 1), we can calculate the gradient vector and then normalize it to obtain a unit vector. The directional derivative is maximized in the direction of this unit vector.

First, let's find the gradient vector of \varphi. The gradient vector is defined as:

`∇\varphi = ({∂\varphi}/{∂x}, {∂\varphi}/{∂y}, {∂\varphi}/{∂z})`

Taking partial derivatives of `\varphi` with respect to each variable, we get:

`{∂\varphi}/{∂x} = y^2z + 2xy`

`{∂\varphi}/{∂y} = 2xyz + x^2`

`{∂\varphi}/{∂z} = xy^2`

Evaluating these partial derivatives at the point (1, 2, 1), we have:

`{∂\varphi}/{∂x} = (2^2)(1) + 2(1)(2) = 8`

`{∂\varphi}/{∂y} = 2(1)(2)(1) + (1)^2 = 5`

`{∂\varphi}/{∂z} = (1)(2^2) = 4`

Therefore, the gradient vector at (1, 2, 1) is:

`∇\varphi = (8, 5, 4)` or  `8\hati + 5\hatj +4\hatk`

Next, For the magnitude of the maximum:

`|∇\varphi| = \sqrt {8^2 + 5^2 + 4^2} = \sqrt {64 + 25 + 16} = \sqrt {105}`