Find the minimum value of the function $f ( x , y , z ) = 2 x + y - 2 z$ subject to the constraint that $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ .
A) $- 4$
B) $- 5$
C) $- 6$
D) $- 7$
E)4
F)5
G)6
H)7

Question

Find the minimum value of the function $f ( x , y , z ) = 2 x + y - 2 z$ subject to the constraint that $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ .
A) $- 4$ 
B) $- 5$
C) $- 6$
D) $- 7$
E)4 
F)5
G)6
H)7

Quizplus · Accepted Answer

Using the method of Lagrange multipliers, we set up the system of equations: $2 = \lambda \cdot 2x$, $1 = \lambda \cdot 2y$, $-2 = \lambda \cdot 2z$, and the constraint $x^2 + y^2 + z^2 = 4$. Solving these equations, we find that the minimum value of $f(x, y, z)$ is $-6$.

Find the Minimum Value of the Function f(x,y,z)=2x+y−2zf ( x , y , z ) = 2 x + y - 2 zf(x,y,z)=2x+y−2z

Find the Minimum Value of the Function $f ( x , y , z ) = 2 x + y - 2 z$