Solve the problem.
-Find the derivative of the function f(x, y, z) = ln(xy + yz + zx) at the point (8, 16, 24) in the direction in which the function increases most rapidly.
A) $\frac { 5 } { 88 } \sqrt { 2 }$
B) $\frac { 5 } { 56 } \sqrt { 2 }$
C) $\frac { 5 } { 104 } \sqrt { 2 }$
D) $\frac { 5 } { 136 } \sqrt { 2 }$

Question

Solve the problem.
-Find the derivative of the function f(x, y, z) = ln(xy + yz + zx) at the point (8, 16, 24) in the direction in which the function increases most rapidly. 
A) $\frac { 5 } { 88 } \sqrt { 2 }$
B) $\frac { 5 } { 56 } \sqrt { 2 }$
C) $\frac { 5 } { 104 } \sqrt { 2 }$
D) $\frac { 5 } { 136 } \sqrt { 2 }$

Quizplus · Accepted Answer

To find the direction of greatest increase, we need to take the gradient vector of the function and normalize it. 
$$
abla f(x,y,z) = \left(\frac{y+z}{xy+yz+zx}, \frac{x+z}{xy+yz+zx}, \frac{x+y}{xy+yz+zx}ight)$$
At the point (8, 16, 24), the gradient vector is 
$$
abla f(8, 16, 24) = \left(\frac{5}{88}, \frac{5}{44}, \frac{5}{22}ight)$$
To normalize, we calculate the magnitude of the gradient vector and divide each component by it. 
$$\left|
abla f(8, 16, 24)ight| = \sqrt{\left(\frac{5}{88}ight)^2 + \left(\frac{5}{44}ight)^2 + \left(\frac{5}{22}ight)^2} = \frac{5}{22\sqrt{2}}$$

Therefore, the direction of greatest increase is 
$$\left(\frac{y+z}{xy+yz+zx}, \frac{x+z}{xy+yz+zx}, \frac{x+y}{xy+yz+zx}ight) = \left(\frac{5}{88\sqrt{2}}, \frac{5}{44\sqrt{2}}, \frac{5}{22\sqrt{2}}ight)$$
Finally, to find the rate of change in the direction of greatest increase, we take the dot product of the normalized gradient vector and the directional vector. The directional vector is 
$$\left(\frac{5}{88\sqrt{2}}, \frac{5}{44\sqrt{2}}, \frac{5}{22\sqrt{2}}ight)$$
and the normalized gradient vector is 
$$
abla f(8, 16, 24) = \left(\frac{5}{88}, \frac{5}{44}, \frac{5}{22}ight)$$
so the dot product is 
$$\frac{5}{22\sqrt{2}} \cdot \frac{5}{88\sqrt{2}} + \frac{5}{22\sqrt{2}} \cdot \frac{5}{44\sqrt{2}} + \frac{5}{22\sqrt{2}} \cdot \frac{5}{22\sqrt{2}} = \frac{5}{88}$$

Therefore, the answer is $\boxed{	extbf{(A) } \frac { 5 } { 88 } \sqrt { 2 }}$.

Solve the Problem 5882\frac { 5 } { 88 } \sqrt { 2 }885​2​

Solve the Problem $\frac { 5 } { 88 } \sqrt { 2 }$