Find the direction of maximum increase of the function $f ( x , y , z ) = x e ^ { - y } + 3 z$ at the point $( 1,0,4 )$ .
A) $\{ 1,1,3 \}$

B) $\{ - 1 , - 1,3 \}$
C) $\{ - 1,1,3 \}$
D) $(1 , - 1,4 )$
E) $\{ 1,3,3 \}$
F) $\{ 1 , - 3,3 \}$

G) $\{ 1 , - 1,3 \}$

H) $\{ - 1,1,4 \}$

Question

Find the direction of maximum increase of the function $f ( x , y , z ) = x e ^ { - y } + 3 z$ at the point $( 1,0,4 )$ .
A) $\{ 1,1,3 \}$

B) $\{ - 1 , - 1,3 \}$ 
C) $\{ - 1,1,3 \}$ 
D) $(1 , - 1,4 )$
E) $\{ 1,3,3 \}$  
F) $\{ 1 , - 3,3 \}$

G) $\{ 1 , - 1,3 \}$

H) $\{ - 1,1,4 \}$

Quizplus · Accepted Answer

The direction of maximum increase of a function is given by its gradient vector. The gradient of $f(x, y, z) = xe^{-y} + 3z$ is $\nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right)$. Calculating the partial derivatives, we get $\nabla f = (e^{-y}, -xe^{-y}, 3)$. Substituting the point $(1,0,4)$ into the gradient, we get $\nabla f(1,0,4) = (1, -1, 3)$, which corresponds to choice G.

Find the Direction of Maximum Increase of the Function f(x,y,z)=xe−y+3zf ( x , y , z ) = x e ^ { - y } + 3 zf(x,y,z)=xe−y+3z

Find the Direction of Maximum Increase of the Function $f ( x , y , z ) = x e ^ { - y } + 3 z$