begin{aligned} & f(x, y)=2(x-y)^2-x^4-y^4 & | vedclass

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(C) Given function is $f(x, y) = 2(x-y)^2 - x^4 - y^4$. First, we find the partial derivatives: $f_x = \frac{\partial}{\partial x} [2(x-y)^2 - x^4 - y

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(C) Given function is $f(x, y) = 2(x-y)^2 - x^4 - y^4$. First, we find the partial derivatives: $f_x = \frac{\partial}{\partial x} [2(x-y)^2 - x^4 - y^4] = 4(x-y) - 4x^3$. $f_y = \frac{\partial}{\partial y} [2(x-y)^2 - x^4 - y^4] = -4(x-y) - 4y^3$. Now, we find the second-order partial derivatives: $f_{xx} = \frac{\partial}{\partial x} [4(x-y) - 4x^3] = 4 - 12x^2$. $f_{yy} = \frac{\partial}{\partial y} [-4(x-y) - 4y^3] = 4 - 12y^2$. $f_{xy} = \frac{\partial}{\partial y} [4(x-y) - 4x^3] = -4$. Evaluating these at $(0,0)$: $f_{xx}(0,0) = 4 - 12(0)^2 = 4$. $f_{yy}(0,0) = 4 - 12(0)^2 = 4$. $f_{xy}(0,0) = -4$. Finally, calculate the expression: $f_{xx}f_{yy} - f_{xy}^2 = (4)(4) - (-4)^2 = 16 - 16 = 0$.

List-$I$	List-$II$
$(1)$ Heavy water	$(a)$ Bicarbonates of $Ca$ and $Mg$ in water
$(2)$ Temporary hard water	$(b)$ No foreign ions in water
$(3)$ Soft water	$(c)$ $D_2O$
$(4)$ Permanent hard water	$(d)$ Sulphates and chlorides of $Ca$ and $Mg$ in water

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

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