Find all the local maxima, local minima, and saddle points of the function.
-$$f ( x , y ) = 81 x ^ { 2 } + 72 x y + 64 y ^ { 2 }$$
A) f(9, 8) = 15,841, saddle point; f(8, 9) = 15,552, saddle point
B) f(72, 72) = 746,496, local maximum; f(0, 0) = 0, local minimum
C) f(0, 0) = 0, local minimum
D) f(72, 72) = 746,496, local maximum

Question

Quizplus · Accepted Answer

To determine the local maxima, local minima, and saddle points of the given function $f(x, y) = 81x^2 + 72xy + 64y^2$, we first find its first partial derivatives and then its second partial derivatives to apply the second derivative test. The first partial derivatives are:$$f_x = 162x + 72y$$$$f_y = 72x + 128y$$Setting these equal to zero gives us the system of equations:$$162x + 72y = 0$$$$72x + 128y = 0$$Solving this system, we find that $x = 0$ and $y = 0$, so there is a critical point at $(0, 0)$.The second partial derivatives are:$$f_{xx} = 162, f_{yy} = 128, f_{xy} = 72$$The determinant of the Hessian matrix at $(0, 0)$ is:$$D = f_{xx}f_{yy} - (f_{xy})^2 = 162 \times 128 - 72^2 = 20736 - 5184 = 15552$$Since $D > 0$ and $f_{xx} > 0$, the function has a local minimum at $(0, 0)$. There are no local maxima or saddle points identified with the given information, making option C the correct answer.

Find All the Local Maxima, Local Minima, and Saddle Points f(x,y)=81x2+72xy+64y2f ( x , y ) = 81 x ^ { 2 } + 72 x y + 64 y ^ { 2 }f(x,y)=81x2+72xy+64y2

Find All the Local Maxima, Local Minima, and Saddle Points $f ( x , y ) = 81 x ^ { 2 } + 72 x y + 64 y ^ { 2 }$