Locate the critical point of the function. $f ( x , y ) = 4 - x ^ { 2 } - 6 x - y ^ { 2 } + 4 y$
A) $( 3,2 , - 17 )$
B) $( - 3,2,17 )$
C) $( 6,4,4 )$
D) $( - 6,4,4 )$
E) $( 2 , - 3,17 )$

Question

Locate the critical point of the function.   $f ( x , y ) = 4 - x ^ { 2 } - 6 x - y ^ { 2 } + 4 y$  
A) $( 3,2 , - 17 )$
B) $( - 3,2,17 )$  
C) $( 6,4,4 )$
D) $( - 6,4,4 )$
E) $( 2 , - 3,17 )$

Quizplus · Accepted Answer

To find the critical point of the function $f(x, y) = 4 - x^2 - 6x - y^2 + 4y$, we first need to find the partial derivatives of $f$ with respect to $x$ and $y$, and then set them equal to zero to solve for $x$ and $y$.The partial derivative of $f$ with respect to $x$ is $\frac{\partial f}{\partial x} = -2x - 6$, and setting it equal to zero gives $x = -3$.The partial derivative of $f$ with respect to $y$ is $\frac{\partial f}{\partial y} = -2y + 4$, and setting it equal to zero gives $y = 2$.Substituting $x = -3$ and $y = 2$ into the original function to find the $z$-value gives $f(-3, 2) = 4 - (-3)^2 - 6(-3) - (2)^2 + 4(2) = 17$.Therefore, the critical point is $(-3, 2, 17)$.

Locate the Critical Point of the Function f(x,y)=4−x2−6x−y2+4yf ( x , y ) = 4 - x ^ { 2 } - 6 x - y ^ { 2 } + 4 yf(x,y)=4−x2−6x−y2+4y

Locate the Critical Point of the Function $f ( x , y ) = 4 - x ^ { 2 } - 6 x - y ^ { 2 } + 4 y$