Final answer:
To determine the extremum values of the function f=2x³ −6xz+2y²−4y+6z², we find and solve its first-order partial derivatives, resulting in critical points that are further analyzed using second derivative tests to identify maxima or minima.
Step-by-step explanation:
To find the extremum (maximum or minimum) values of the function f=2x³ −6xz+2y²−4y+6z², we need to calculate its first-order partial derivatives and set them to zero to find critical points. These critical points are then analyzed using the second-order partial derivatives to determine if they are maxima, minima, or saddle points through the second derivative test.
The first-order partial derivatives are:
- f'_x = 6x² - 6z
- f'_y = 4y - 4
- f'_z = -6x + 12z
Setting these derivatives equal to zero gives us a system of equations. Solving these equations will give us the critical points:
- 6x² - 6z = 0
- 4y - 4 = 0
- -6x + 12z = 0
After solving these equations, we find the potential extreme points to which we will apply the second derivative test based on the function's Hessian matrix to determine the nature of these points.