Provide an appropriate response.
-Find the direction in which the function is increasing most rapidly at the point $\mathrm { P } _ { 0 }$. $f ( x , y , z ) = x y - \ln ( z ) , P _ { 0 } ( 1,2,2 )$
A) $\frac { 1 } { 21 } ( 4 \mathrm { i } + 2 \mathrm { j } - \mathrm { k } )$
B) $\frac { 1 } { \sqrt { 21 } } ( 4 i - 2 j + k )$
C) $\frac { 21 } { \sqrt { 21 } } ( 4 i + 2 j - k )$
D) $\frac { 1 } { \sqrt { 21 } } ( 4 i + 2 j - k )$

Question

Quizplus · Accepted Answer

The direction in which the function $f(x, y, z) = xy - \ln(z)$ increases most rapidly at a point is given by the gradient of $f$ at that point, $\nabla f$. The gradient is calculated as $\nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right)$. For $f(x, y, z) = xy - \ln(z)$, we have $\frac{\partial f}{\partial x} = y$, $\frac{\partial f}{\partial y} = x$, and $\frac{\partial f}{\partial z} = -\frac{1}{z}$. At $P_0(1,2,2)$, $\nabla f = (2, 1, -\frac{1}{2})$. The direction of maximum increase is the unit vector in the direction of $\nabla f$, which is $\frac{1}{\sqrt{4 + 1 + \frac{1}{4}}} (2i + j - \frac{1}{2}k) = \frac{1}{\sqrt{21}} (4i + 2j - k)$.

Provide an Appropriate Response P0\mathrm { P } _ { 0 }P0​

Provide an Appropriate Response $\mathrm { P } _ { 0 }$