Final answer:
The equation for the tangent plane to the 9-level surface of f(x, y, z) = 5x² + 3y² + z² at P0(1, 1, 1) is 10x + 6y + 2z = 18.
Step-by-step explanation:
To find the equation for the tangent plane to the 9-level surface of f(x, y, z) = 5x² + 3y² + z² at P0(1, 1, 1), we need to calculate the gradient of f at P0. The gradient, denoted as ∇f(P0), is composed of the partial derivatives of f with respect to x, y, and z evaluated at P0. The partial derivatives are:
∂f/∂x = 10x
∂f/∂y = 6y
∂f/∂z = 2z
Evaluating these at P0 gives us ∇f(P0) = <10, 6, 2>. The equation of the tangent plane at P0 can then be written as:
10(x - 1) + 6(y - 1) + 2(z - 1) = 0
Simplifying this equation yields:
10x + 6y + 2z = 18