Mathematics-I (3110014)

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

$\mathrm{Now},$

$\frac{\partial \mathrm{f}}{\partial \mathrm{x}}=\frac{\partial }{\partial \mathrm{x}}\left(3{\mathrm{x}}^2-{\mathrm{y}}^2+{\mathrm{x}}^3\right)$

$\frac{\partial \mathrm{f}}{\partial \mathrm{x}}=\frac{\partial }{\partial \mathrm{x}}\left(3{\mathrm{x}}^2\right)-\frac{\partial }{\partial \mathrm{x}}\left({\mathrm{y}}^2\right)+\frac{\partial }{\partial \mathrm{x}}\left({\mathrm{x}}^3\right)$

$\frac{\partial \mathrm{f}}{\partial \mathrm{x}}=6\mathrm{x}-0+3{\mathrm{x}}^2$

$\frac{\partial \mathrm{f}}{\partial \mathrm{x}}=6\mathrm{x}+3{\mathrm{x}}^2$

$\frac{\partial \mathrm{f}}{\partial \mathrm{y}}=\frac{\partial }{\partial \mathrm{y}}\left(3{\mathrm{x}}^2-{\mathrm{y}}^2+{\mathrm{x}}^3\right)$

$\frac{\partial \mathrm{f}}{\partial \mathrm{y}}=\frac{\partial }{\partial \mathrm{y}}\left(3{\mathrm{x}}^2\right)-\frac{\partial }{\partial \mathrm{y}}\left({\mathrm{y}}^2\right)+\frac{\partial }{\partial \mathrm{y}}\left({\mathrm{x}}^3\right)$

$\frac{\partial \mathrm{f}}{\partial \mathrm{y}}=0-2y+0$

$\frac{\partial \mathrm{f}}{\partial \mathrm{y}}=-2y$

$\mathrm{For}, \mathrm{Stationary} \mathrm{point},$

$\frac{\partial \mathrm{f}}{\partial \mathrm{x}}=0$

$⟹6\mathrm{x}+3{\mathrm{x}}^2=0$

$⟹3\mathrm{x}(2+\mathrm{x})=0$

$⟹3\mathrm{x}=0 \mathrm{OR} 2+\mathrm{x}=0$

$⟹\mathrm{x}=0 \mathrm{OR} \mathrm{x}=-2$

$\frac{\partial \mathrm{f}}{\partial \mathrm{y}}=0$

$⟹-2\mathrm{y}=0$

$⟹\mathrm{y}=0$

$\mathrm{Hence}, \mathrm{Stationary} \mathrm{points} \mathrm{are} (0,0) \& (-2,0).$

$\mathrm{Now},$

$\mathrm{r}=\frac{{\partial }^2\mathrm{f}}{\partial {\mathrm{x}}^2}$

$\mathrm{r}=\frac{\partial }{\partial \mathrm{x}}\left(\frac{\partial \mathrm{f}}{\partial \mathrm{x}}\right)$

$\mathrm{r}=\frac{\partial }{\partial \mathrm{x}}\left(6\mathrm{x}+3{\mathrm{x}}^2\right)$

$\mathrm{r}=\frac{\partial }{\partial \mathrm{x}}\left(6\mathrm{x}\right)+\frac{\partial }{\partial \mathrm{x}}\left(3{\mathrm{x}}^2\right)$

$\mathrm{r}=6+6\mathrm{x}$

$\mathrm{s}=\frac{{\partial }^2\mathrm{f}}{\partial \mathrm{x}\partial \mathrm{y}}$

$\mathrm{s}=\frac{\partial }{\partial \mathrm{x}}\left(\frac{\partial \mathrm{f}}{\partial \mathrm{y}}\right)$

$\mathrm{s}=\frac{\partial }{\partial \mathrm{x}}\left(-2\mathrm{y}\right)$

$\mathrm{s}=0$

$\mathrm{t}=\frac{{\partial }^2\mathrm{f}}{\partial {\mathrm{y}}^2}$

$\mathrm{t}=\frac{\partial }{\partial \mathrm{y}}\left(\frac{\partial \mathrm{f}}{\partial \mathrm{y}}\right)$

$\mathrm{s}=\frac{\partial }{\partial \mathrm{y}}\left(-2\mathrm{y}\right)$

$\mathrm{s}=-2$