Let $f ( x , y ) = x y ^ { 2 } + x ^ { 2 }$ . Then the maximum value of the directional derivative $D _ { \mathrm { u } } f ( - 1,2 )$ is

A) $\sqrt { 5 }$
B) $2 \sqrt { 3 }$
C) $2 \sqrt { 5 }$
D) $\sqrt { 35 }$
E)5

Question

Let $f ( x , y ) = x y ^ { 2 } + x ^ { 2 }$ . Then the maximum value of the directional derivative $D _ { \mathrm { u } } f ( - 1,2 )$ is

A) $\sqrt { 5 }$
B) $2 \sqrt { 3 }$
C) $2 \sqrt { 5 }$
D) $\sqrt { 35 }$
E)5

Quizplus · Accepted Answer

The maximum value of the directional derivative is the magnitude of the gradient vector at the point. The gradient of $f$ is $\nabla f = (y^2 + 2x, 2xy)$. At $(-1, 2)$, $\nabla f = (4 - 2, -4) = (2, -4)$. The magnitude is $\sqrt{2^2 + (-4)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5}$.

Let f(x,y)=xy2+x2f ( x , y ) = x y ^ { 2 } + x ^ { 2 }f(x,y)=xy2+x2

Let $f ( x , y ) = x y ^ { 2 } + x ^ { 2 }$