Find the maximum 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 maximum 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

To find the maximum value of $f(x, y, z) = 2x + y - 2z$ subject to $x^2 + y^2 + z^2 = 4$, we can use the method of Lagrange multipliers. However, a quicker approach here is to notice that the constraint describes a sphere with radius 2. The function $f(x, y, z)$ can be interpreted as a dot product between $(2, 1, -2)$ and $(x, y, z)$. The maximum value occurs when $(x, y, z)$ points in the same direction as $(2, 1, -2)$, and the magnitude of $(x, y, z)$ is the radius of the sphere, which is 2. Thus, the maximum value is the magnitude of $(2, 1, -2)$ times the radius. The magnitude of $(2, 1, -2)$ is $\sqrt{2^2 + 1^2 + (-2)^2} = \sqrt{9} = 3$, so the maximum value is $3 \times 2 = 6$.

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