Final answer:
To find the equation of the tangent plane to the level surface of the function at the given point, we compute the gradient vector at that point and use it in the general equation of the tangent plane, resulting in the equation 10x - 6y + 2z = 6.
Step-by-step explanation:
To find an equation for the tangent plane to the 9-level surface of f(x, y, z) = 5x² - 3y² + z² at the point p0(1, 1, 1), we first need to compute the gradient of f at p0.
The gradient of f is the vector of partial derivatives (∂f/∂x, ∂f/∂y, ∂f/∂z).
At p0(1, 1, 1), the partial derivatives are 10, -6, and 2 respectively.
Therefore, the gradient at p0 is (10, -6, 2).
The equation of the tangent plane at p0 is given by (x - 1)∗10 + (y - 1)∗(-6) + (z - 1)∗2 = 0, which simplifies to 10x - 6y + 2z = 6.