Find the maximum rate of change of $f ( x , y ) = x y ^ { 2 } + \sqrt { y }$ at the point $( 2,1 )$. In what direction does it occur?
A) $\frac { 85 } { 2 } , \left( 2 , \frac { 7 } { 2 } \right)$
B) $\frac { 29 } { 2 } , \left( 1 , \frac { 9 } { 2 } \right)$
C) $\frac { \sqrt { 29 } } { 2 } , \left( 1 , \frac { 5 } { 2 } \right)$
D) $\frac { \sqrt { 85 } } { 2 } , \left( 1 , \frac { 9 } { 2 } \right)$
E) none of these

Question

Quizplus · Accepted Answer

The maximum rate of change is the magnitude of the gradient vector at the point (2,1). The gradient is $\nabla f = (y^2, 2xy + \frac{1}{2\sqrt{y}})$. At (2,1), $\nabla f = (1, 5)$. The magnitude is $\sqrt{1^2 + 5^2} = \sqrt{26}$. The direction is the unit vector $\left(\frac{1}{\sqrt{26}}, \frac{5}{\sqrt{26}}\right)$, which simplifies to $\left(1, \frac{9}{2}\right)$ after scaling.

Find the Maximum Rate of Change Of f(x,y)=xy2+yf ( x , y ) = x y ^ { 2 } + \sqrt { y }f(x,y)=xy2+y​

Find the Maximum Rate of Change Of $f ( x , y ) = x y ^ { 2 } + \sqrt { y }$