Exam 14: Directional Derivatives, Gradients, and Extrema

The absolute maximum of $f ( x , y ) = 88 x + 114 y - 2 x ^ { 2 } - 3 y ^ { 2 } - 4 x y$ on the set $\left\{ ( x , y ) : 0 \leq x \leq 50,0 \leq y \leq \frac { 125 } { 3 } \right\}$ is

Let $f ( x , y ) = x ^ { 3 } + y ^ { 3 } + 3 y ^ { 2 } - 3 x - 9 y + 2$ . Then f has a relative maximum at

Let $f ( x , y ) = 4 x y ^ { 2 } - 2 x ^ { 2 } y - x$ . Then the set of saddle point(s) of f is

The gradient of $f ( x , y , z ) = 2 x ^ { 3 } + x y ^ { 2 } + z ^ { 2 } x$ at (1, 1, 1) is

Let $w = ( x - 1 ) ^ { 2 } + ( y - 3 ) ^ { 2 } + z ^ { 2 }$ with constraint $4 x + 2 y - z = 5$ . Then the minimum value of w is

Let $f ( x , y ) = x ^ { 2 } y + y ^ { 2 }$ and $\mathbf { u } = \frac { 1 } { 2 } \mathbf { i } + \frac { \sqrt { 3 } } { 2 } \mathbf { j }$ Then the directional derivative of f at (-1, 2) in the direction of $\mathbf { U }$ is

Let $w = ( x - 1 ) ^ { 2 } + ( y + 1 ) ^ { 2 } + ( z + 1 ) ^ { 2 }$ with constraint $x + 4 y + 3 z = 2$ . Then the minimum value of w is

Let $f ( x , y ) = \ln \sqrt { x ^ { 2 } + y ^ { 2 } }$ . Then the directional derivative of f at (3,4) in the direction of the unit vector $\mathbf { U }$ which is parallel to $\mathbf { v } = 5 \mathbf { i } + 12 \mathbf { j }$ is

The gradient of $f ( x , y , z ) = x ^ { 2 } y - x y z ^ { 2 }$ at (0, 1, 2) is

Let $w = x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$ with constraint $x + y + z = 1$ . Then the maximum value of w is

Let $f ( x , y ) = x ^ { 2 } + y$ with constraint $x ^ { 2 } + y ^ { 2 } = 9$ . Then f has a relative minimum at

An equation of the tangent plane to the surface $3 z = x ^ { 2 } + y ^ { 2 } - 2$ at (-2, -4, 6) is

The symmetric equations of the normal line to the surface $y = e ^ { x } \cos z$ at (1, e, 0) are

Let $f ( x , y , z ) = x ^ { 2 } y + y ^ { 2 } z + z ^ { 2 } x$ Then the directional derivative of f at $P = ( 1,2 , - 1 )$ in the direction of the unit vector $\mathbf { u }$ which is parallel to $\overrightarrow { P Q }$ where $Q = ( 2,0,1 )$ is

Let $f ( x , y , z ) = x y z$ with constraint $2 x y + 3 x z + y z = 72$ . Then f has a relative maximum at

The absolute minimum of $f ( x , y ) = 2 x + 2 y - x ^ { 2 } - y ^ { 2 } + 2$ on or inside the triangle with vertices (0, 0), (9, 0), and (0,9) is

Let $f ( x , y ) = \frac { 2 x + 2 y + 1 } { x ^ { 2 } + y ^ { 2 } + 1 }$ . Then f has a relative minimum at

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

Let $f ( x , y , z ) = x ^ { 2 } + 4 y ^ { 2 } + 16 z ^ { 2 }$ with constraint xy = 1. Then f has a relative minimum at

The gradient of $f ( x , y , z ) = \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } }$ at (3, 0, -4) is