Answer: We have $f(x,y)=x^3+xy^2-12x^2-2y^2+21x+10.$

For stationary points, $p=\dfrac{\partial f}{\partial x}=0, q=\dfrac{\partial f}{\partial y}=0$.

$p=\dfrac{\partial f}{\partial x}=3x^2+y^2-24x+21=0$......................................(1)

$q=\dfrac{\partial f}{\partial y}=2xy-4y=0 \Rightarrow 2y(x-2)=0$............................(2)

From (2), either $y=0$ or $x=2.$

Putting $y=0$ in (1), we get

 $3x^2-24x+21=0 \Rightarrow x^2-8x+7=0$

$x^2-x-7x+7=0 \Rightarrow x(x-1)-7(x-1)=0. $

$\Rightarrow (x-7)(x-1)=0 \Rightarrow x=1,7$

So, two stationary points are $(1,0)$ and $(7,0).$

Similarly, putting $x = 2$ in (1), we get 

$12+y^2-48+21=0 \Rightarrow y^2-15=0 \Rightarrow y=\sqrt{15}, -\sqrt{15}.$

Thus, two more stationary points are $(2, \sqrt{15}), (2, -\sqrt{15}).$

We have four stationary points $(1,0), (7,0), (2, \sqrt{15}), (2, -\sqrt{15}).$

For the analysis of maxima and minima of the points, we have to find $r=\dfrac{\partial^2 f}{\partial x^2}, s=\dfrac{\partial ^2 f}{\partial x \partial y}, t=\dfrac {\partial^2 f}{\partial y^2}, rt-s^2=\dfrac{\partial^2 f}{\partial x^2}\dfrac {\partial^2 f}{\partial y^2}-(\dfrac{\partial ^2 f}{\partial x \partial y})^2$at the above four points.

$r=\dfrac{\partial ^2 f}{\partial x^2}=6x-24,$      $ s=\dfrac{\partial ^2 f}{\partial x \partial y}=2y,$    $ t=\dfrac {\partial^2 f}{\partial y^2}=2x-4.$

$0$ $-2$ \(36\gt0\) | $18$ $0$ $10$ (180>0) | $-12$ $2 \sqrt{15}$ $0$ \(-60\lt0\) | $-12$ $-2 \sqrt{15}$ $0$ \(-60\lt0\) |  As \(rt-s^2\lt0\) for $ (2, \sqrt{15})$ and $(2, -\sqrt{15})$, hence there is no extrema (Maximum/ Minimum value). For $(1,0)$ and $(7,0)$ \(rt-s^2\gt0\), hence it has extremum value. **For $(1,0)$,** \(rt-s^2\gt0\) and \(r\lt0,t\lt0,\) hence it has maximum value and the maximum value is $f(1,0)=1+0-12-0+21+10=30.$ **For $(7,0),$ \(rt-s^2\gt0\)**and \(r\gt0,t\gt0,\) hence it has minimum value and the minimum value is $f(7,0)=7^3-12 \times7^2+21 \times7=343-588+147=-98.$ **ThusStationary Points are : $(1,0), (7,0),(2, \sqrt{15}), (2, -\sqrt{15})$** Maximum at $(1,0)$ value is $30.$ Minimum at $(7,0)$ value is $-98.$