GATE (TF) Textile 2021 Question Paper Solution | GATE/2021/TF/27

If the function $f(x,y)$ is defined by
$f(x,y)=x^3 - \frac{3}{2}x^2y^2+y^3, \ \ \ \ \ \ \ \ x,y$ $\epsilon$ $R$
then

(A)	Neither (0, 0) nor (1, 1) is a critical point
(B)	(0, 0) is a critical point but (1, 1) is NOT a critical point
(C)	(0,0) is NOT a critical point but (1,1) is a critical point
()	(0,0) and (0.1) are both critical points

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Given

$f(x,y)=x^3 - \frac{3}{2}x^2y^2+y^3$

$\frac{\delta f}{\delta x}=3x^2-\frac{3}{2}\times 2x \times y^2$

$\frac{\delta f}{\delta x}=3x^2-3\times x \times y^2$ =0

Then , $x=y^2$

Similarily-

$\frac{\delta f}{\delta y}=-3\times x^2 \times y+3y^2$ =0 ,then, x=y

Critical points-

$D=\frac{\delta^2f}{\delta x^2} \times \frac{\delta^2f}{\delta y^2}-(\frac{\delta^2f}{\delta x \times \delta y})^2<0$

(0,0) and (1,1) both are critical points