MATH SOLVE

5 months ago

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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 = 17Solve for z in the constraint:z = (4x + 3y − 17) / 3Substitute:f(x,y) = x² + y² + (4x + 3y − 17)² / 9Find 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 − 1360 = 50x + 24y − 1360 = 25x + 12y − 680 = 2y + 2/3 (4x + 3y − 17)0 = 6y + 2 (4x + 3y − 17)0 = 6y + 8x + 6y − 340 = 8x + 12y − 34Solve the system of equations.0 = (25x + 12y − 68) − (8x + 12y − 34)0 = 17x − 34x = 2y = 1.5Solve for z:z = (4x + 3y − 17) / 3z = -1.5Evaluate the function at the point:f(2,1.5,-1.5) = (2)² + (1.5)² + (-1.5)²f(2,1.5,-1.5) = 8.5There 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 > 0Both 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 / 9f(0,0) = 0² + 0² + (0 + 0 − 17)² / 9f(0,0) = 32.111This is greater than f(2,1.5,-1.5), so f(2,1.5,-1.5) must be a minimum.