Let $f ( x , y , z ) = 3 \sqrt { x ^ { 2 } + 4 y ^ { 2 } + z ^ { 2 } }$ . Find $f _ { z } ( 2,1,1 )$ .
A)0
B)1
C) $\frac { 1 } { 2 }$
D)3
E)4
F) $- 1$
G) $- \frac { 1 } { 2 }$
H) $- 3$

Question

Let $f ( x , y , z ) = 3 \sqrt { x ^ { 2 } + 4 y ^ { 2 } + z ^ { 2 } }$ . Find $f _ { z } ( 2,1,1 )$ .
A)0
B)1
C) $\frac { 1 } { 2 }$ 
D)3
E)4
F) $- 1$ 
G) $- \frac { 1 } { 2 }$
H) $- 3$

Quizplus · Accepted Answer

To find $f_z(2,1,1)$, we need to compute the partial derivative of $f$ with respect to $z$ and then evaluate it at the point $(2,1,1)$. The partial derivative of $f$ with respect to $z$ is given by $\frac{\partial f}{\partial z} = \frac{3 \cdot 2z}{2\sqrt{x^2 + 4y^2 + z^2}}$. Substituting $x=2$, $y=1$, and $z=1$ into this expression gives $\frac{3 \cdot 2 \cdot 1}{2\sqrt{2^2 + 4\cdot1^2 + 1^2}} = \frac{3}{\sqrt{4 + 4 + 1}} = \frac{3}{3} = 1$.

Let f(x,y,z)=3x2+4y2+z2f ( x , y , z ) = 3 \sqrt { x ^ { 2 } + 4 y ^ { 2 } + z ^ { 2 } }f(x,y,z)=3x2+4y2+z2​

Let $f ( x , y , z ) = 3 \sqrt { x ^ { 2 } + 4 y ^ { 2 } + z ^ { 2 } }$