Given the function defined above, The correct option is (Option C). That is, there is only one solution where fx(x,y)=0 and fy(x,y)=0, when x=enter your response here and y=enter your response here.(Type integers or simplified fractions.)
fx(x,y) = 2x² + 9y² + 6xy + 36x-6
fx (x,y) = 0
⇒ 4x + 6y + 36 = 0 ......................1
fy (x,y) = 0
⇒ 18y + 6x = 0
⇒ 6x + 18 y = 0 .............................2
From Eqn 1, 2x + 3y = -18
From Equn 2, x + 3y = 0
Subtract, x = -18
Hence; 3y + x = 0
⇒ 3y - 18 = 0
⇒ y = 6
Hence, it is right to state that Option C is the right option if Fx = 0; and Fy = 0
[tex]\boxed{X= -18; Y = 6}[/tex]
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Full Question:
For the function defined as follows, find all values of x and y such that both fx(x,y)=0 and fy(x,y)=0. f(x,y)=2x^2+9y^2+6xy+36x−6
Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
A. There are two solutions where fx(x,y)=0 and fy(x,y)=0, in order from increasing x values, when x=enter your response here and y=enter your response here and x=enter your response here and y=enter your response here. (Type integers or simplifiedfractions.)
B. There are three solutions where fx(x,y)=0 and fy(x,y)=0, in order from increasing x values, when x=enter your response here and y=enter your response here and x=enter your response here and y=enter your response here and x=enter your response here and y=enter your response here. (Type integers or simplifiedfractions.)
C. There is only one solution where fx(x,y)=0 and fy(x,y)=0, when x=enter your response here and y=enter your response here.(Type integers or simplified fractions.)
D. There are no solutions where fx(x,y)=0 and fy(x,y)=0.
