Answer:
There is a minimum value of 8.5 located at (x,y,z) = (2, 1.5, -1.5).
Step-by-step explanation:
f(x,y,z) = x² + y² + z², 4x + 3y − 3z = 17
Solve for z in the constraint:
z = (4x + 3y − 17) / 3
Substitute:
f(x,y) = x² + y² + (4x + 3y − 17)² / 9
Find the partial derivatives:
∂f/∂x = 2x + 2(4x + 3y − 17) / 9 (4)
∂f/∂x = 2x + 8/9 (4x + 3y − 17)
∂f/∂y = 2y + 2(4x + 3y − 17) / 9 (3)
∂f/∂y = 2y + 2/3 (4x + 3y − 17)
Set the partial derivatives to 0:
0 = 2x + 8/9 (4x + 3y − 17)
0 = 18x + 8 (4x + 3y − 17)
0 = 18x + 32x + 24y − 136
0 = 50x + 24y − 136
0 = 25x + 12y − 68
0 = 2y + 2/3 (4x + 3y − 17)
0 = 6y + 2 (4x + 3y − 17)
0 = 6y + 8x + 6y − 34
0 = 8x + 12y − 34
Solve the system of equations.
0 = (25x + 12y − 68) − (8x + 12y − 34)
0 = 17x − 34
x = 2
y = 1.5
Solve for z:
z = (4x + 3y − 17) / 3
z = -1.5
Evaluate the function at the point:
f(2,1.5,-1.5) = (2)² + (1.5)² + (-1.5)²
f(2,1.5,-1.5) = 8.5
There are a number of ways to determine whether this is a minimum or maximum. One is by finding the second partial derivatives and evaluating at the point.
∂²f/∂x² = 50/9 > 0
∂²f/∂y² = 4 > 0
Both are positive, so the extremum is a minimum.
Another way is by simply evaluating the function at a different point and comparing.
f(x,y) = x² + y² + (4x + 3y − 17)² / 9
f(0,0) = 0² + 0² + (0 + 0 − 17)² / 9
f(0,0) = 32.111
This is greater than f(2,1.5,-1.5), so f(2,1.5,-1.5) must be a minimum.
