Question

If $f(x,y)=-10x^2+6xy-6x-y^2-1$ where $x,y\in R$ is a quadratic polynomial in two variable, then which of the following options is/are true?

• A. The maximum value of $f(x,y)$ is $8$
• B. The minimum value of $f(x,y)$ is $8$
• C. The maximum value of $f(x,y)$ is occurs at $x=-3$ and $y=-9$
• D. The maximum value of $f(x,y)$ is occurs at $x=-3$ and $y=9$

Answer 1

Answer: (A), (C)

The correct options are
A The maximum value of $f(x,y)$ is $8$
C The maximum value of $f(x,y)$ is occurs at $x=-3$ and $y=-9$

$f(x,y)=-10x^2+6xy-6x-y^2-1=-(9x^2-6xy+y^2)-(x^2+6x+9)+8=-(3x-y{)}^2-(x+3{)}^2+8\Rightarrow f(x,y)=8-\left[\right(3x-y{)}^2+(x+3{)}^2]$

As
$(3x-y{)}^2+(x+3{)}^2\ge 0\Rightarrow 8-\left[\right(3x-y{)}^2+(x+3{)}^2]\le 8\therefore f(x,y)\le 8$
The maximum value of $f(x,y)=8$, when
$x+3=0,3x-y=0\Rightarrow x=-3,y=-9$