Question

Find f xxz ​ (x,y,z) for f(x,y,z)= 3x 2 +6y 2 +z 2 ​ . Provide your answer below: f xxz ​ (x,y,z)=

Answer 1

The second mixed partial derivative
`\(f_(xxz)\)` of the given function
`\(f(x, y, z) = 3x^2 + 6y^2 + z^2\)` is 0.

Step-by-step explanation:

The second mixed partial derivative
`\(f_(xxz)\)` of the given function
`\(f(x, y, z) = 3x^2 + 6y^2 + z^2\)` is calculated by taking the partial derivative of
`\(f_x\)` with respect to z, where
`\(f_x\)` is the partial derivative of f with respect to x. Let's find
`\(f_x\)` first:

`\[ f_x = (\partial f)/(\partial x) = 6x \]`

Now, take the partial derivative of
`\(f_x\)` with respect to z to get
`\(f_(xxz)\)`:

`\[ f_(xxz) = (\partial f_x)/(\partial z) \]`

Since
`\(f_x = 6x\),` the partial derivative of
`\(f_x\)` with respect to z is 0, as there is no z-dependency in
`\(f_x\)`. Therefore,
`\(f_(xxz) = 0\).`

In conclusion, the second mixed partial derivative
`\(f_(xxz)\)` for the given function is 0. This implies that the rate of change of the rate of change of f with respect to x and then z is zero. It indicates that the order in which the partial derivatives with respect to x and z are taken does not affect the result in this specific case, and the function exhibits no variation concerning these specific variables.