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Answer 1

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.