For the function $z=32-3 x^{2}-4 y^{2}$, find each of the following: a. Equation of the tangent plane at $(2,1,16)$ b. A vector that is perpendicular to the tangent plane at the point $(2,1,16)$ c. Points where the tangent plane is horizontal?

Step 1 ：Find the partial derivatives of the function with respect to x and y:

Step 2 ：\(\frac{\partial z}{\partial x} = -6x\)

Step 3 ：\(\frac{\partial z}{\partial y} = -8y\)

Step 4 ：Evaluate the partial derivatives at the point (2,1,16) to find the normal vector:

Step 5 ：\(\vec{n} = (-12, -8, 1)\)

Step 6 ：The equation of the tangent plane is:

Step 7 ：\(-12x - 8y + z - 4 = 0\)

Step 8 ：A vector perpendicular to the tangent plane is:

Step 9 ：\(\vec{v} = (-12, -8, 1)\)

Step 10 ：The tangent plane is horizontal at all points in the xy-plane.