Q111 / 160 · TS EAMCET 2004 · Mathematics

$\begin{aligned} & f(x, y)=2(x-y)^2-x^4-y^4 \\ & \left|\left(f_{x x} f_{y y}-f_{x y}^2\right)\right|_{(0,0)}\end{aligned}$

  • A
    $32$
  • B
    $16$
  • Correct answer
    C
    $0$
  • D
    $-1$

Solution

Given that, $$ f(x, y)=2(x-y)^2-x^4-y^4, \text { we get } $$ On differentiating partially w.r.t. $x$, $$ f_x=4(x-y)-4 x^3 $$ Again differentiating partially, we get $$ \begin{aligned} & & f_{x x} & =4-12 x^2 \\ \Rightarrow & & \left(f_{x x}\right)_{(0,0)} & =4-0=4 \\ \text { Similarly } & & f_{y y} & =4-12 y^2 \\ \Rightarrow & & \left(f_{y y}\right)_{(0,0)} & =4-0=4 \\ \text { and } & & f_{x y} & =-4+0 \\ \Rightarrow & & \left(f_{x y}\right)_{(0,0)} & =-4 \\ & \therefore & \left(f_{x x} f_{y y}-f_{x y}^2\right)_{(0,0)} & =4(4)-(-4)^2=0 \end{aligned} $$

Hence, the correct answer is option: 'C'.

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