Find the critical points of the function. $f ( x , y ) = 72 + 684 x y + 76 x ^ { 2 } + 2160 y + \frac { 9 y ^ { 4 } } { 4 }$
A) $( - 4,4 ) , ( - 6,6 ) , ( 10 , - 10 )$
B) $( - 4,6 ) , ( - 8,6 ) , ( 8 , - 6 )$
C) $( - 4,6 ) , ( - 6 , - 10 ) , ( 10,4 )$
D) $( - 6,6 ) , ( - 8,8 ) , ( 8 , - 8 )$
E) $( 4,4 ) , ( 6,6 ) , ( - 10,0 )$

Question

Quizplus · Accepted Answer

To find the critical points, we need to find where the partial derivatives are equal to zero.$$\frac{\partial f}{\partial x} = 684y + 152x = 0$$$$\frac{\partial f}{\partial y} = 684x + 9y^3 + 2160 = 0$$Solving these equations simultaneously, we get the critical points:$$(-4, 6), (-6, -10), (10, 4)$$

Find the Critical Points of the Function f(x,y)=72+684xy+76x2+2160y+9y44f ( x , y ) = 72 + 684 x y + 76 x ^ { 2 } + 2160 y + \frac { 9 y ^ { 4 } } { 4 }f(x,y)=72+684xy+76x2+2160y+49y4​

Find the Critical Points of the Function $f ( x , y ) = 72 + 684 x y + 76 x ^ { 2 } + 2160 y + \frac { 9 y ^ { 4 } } { 4 }$