Find all the local maxima, local minima, and saddle points of the function.
-$$f ( x , y ) = x ^ { 2 } + 7 x y + y ^ { 2 }$$
A) f(7, 7) = 441, local maximim
B) f(7, 0) = 49, local minimum; f(0, 7) = 49, local minimum
C) f(0, 0) = 0, saddle point; f(7, 7) = 441, local maximim
D) f(0, 0) = 0, saddle point

Question

Quizplus · Accepted Answer

To determine the local maxima, minima, and saddle points of the function $f(x, y) = x^2 + 7xy + y^2$, we first find its critical points by setting its partial derivatives equal to zero. The partial derivatives are:$$f_x = 2x + 7y$$$$f_y = 7x + 2y$$Setting $f_x = 0$ and $f_y = 0$ gives us a system of linear equations:$$2x + 7y = 0$$$$7x + 2y = 0$$Solving this system yields $x = 0$ and $y = 0$, so the only critical point is at $(0, 0)$.To classify this critical point, we use the second derivative test, which involves the Hessian matrix $H$:$$H = \begin{bmatrix} f_{xx} & f_{xy} \ f_{yx} & f_{yy} \end{bmatrix} = \begin{bmatrix} 2 & 7 \ 7 & 2 \end{bmatrix}$$The determinant of $H$ at $(0, 0)$ is $D = f_{xx}f_{yy} - (f_{xy})^2 = (2)(2) - (7)^2 = -45$.Since $D < 0$, the critical point $(0, 0)$ is a saddle point. There are no local maxima or minima because there are no other critical points and the determinant of the Hessian matrix at the critical point indicates a saddle point, not a maximum or minimum.

Find All the Local Maxima, Local Minima, and Saddle Points f(x,y)=x2+7xy+y2f ( x , y ) = x ^ { 2 } + 7 x y + y ^ { 2 }f(x,y)=x2+7xy+y2

Find All the Local Maxima, Local Minima, and Saddle Points $f ( x , y ) = x ^ { 2 } + 7 x y + y ^ { 2 }$