So, I have $R^3$ plane and I also have $F(x,y,z)=100-x^2-yz$.
I want to know the speed of change for $F(x,y,z)$ at the point $(1,2,3)$ and what is the minimum growth for the point $(1,2,3)$ (and direction).
Now, I know that I probably should start by partially derivating $F$, but that's the easy part that is not hard at all, but I'm not sure how to carry on after that.
$$F(x,y,z)=100-x^2-yz$$
The mimimum growth direction is along $-\nabla F$.
$$\nabla F(x,y,z)=\begin{bmatrix} -2x \\-z \\ -y \end{bmatrix}.$$
$$\nabla F(1,2,3)=\begin{bmatrix} -2 \\-3 \\ -2 \end{bmatrix}.$$
Hence the minimum growth rate is $$\frac{-\langle \nabla F(1,2,3), \nabla (1,2,3) \rangle}{\|\nabla F(1,2,3)\|}=-\|\nabla F(1,2,3)\|=-\sqrt{2^2+3^2+2^2}=-\sqrt{17}$$