91Ó°ÊÓ

The function is$g\left(s,t\right)=s^{2}t{e}^{-s{t}^2}$.

The discriminant is calculated by formula,

$det\left(H_g\right)=\frac{{∂}^{2}g}{∂s^2}·\frac{{∂}^{2}g}{∂t^2}-{\left(\frac{{∂}^{2}g}{∂s∂t}\right)}^2$.

calculate $\frac{∂g}{∂s}$, role="math" localid="1649937356777" $\frac{∂g}{∂t}$, $\frac{{∂}^{2}g}{∂s^2}$, role="math" localid="1649937429071" $\frac{{∂}^{2}g}{∂t^2}$and role="math" localid="1649937557097" $\frac{{∂}^{2}g}{∂s∂t}$

$\frac{∂g}{∂s}=2st{e}^{-s{t}^2}-s^2{t}^3{e}^{-s{t}^2}$, role="math" localid="1649937696424" $\frac{{∂}^{2}g}{∂s^2}=2t{e}^{-s{t}^2}-4s{t}^3{e}^{-s{t}^2}+s^2{t}^5{e}^{-s{t}^2}$

$\frac{∂g}{∂t}=s^2{e}^{-s{t}^2}-2s^3{t}^2{e}^{-s{t}^2}$

role="math" localid="1649937738578" $\frac{{∂}^{2}g}{∂t^2}=-6s^{3}t{e}^{-s{t}^2}+4s^4{t}^3{e}^{-s{t}^2}$

role="math" localid="1649937776674" $\frac{{∂}^{2}g}{∂s∂t}=2s{e}^{-s{t}^2}-2s^2{t}^2{e}^{-s{t}^2}+2s^3{t}^4{e}^{-s{t}^2}$

Calculate $det\left(H_g\right)=\frac{{∂}^{2}g}{∂s^2}·\frac{{∂}^{2}g}{∂t^2}-{\left(\frac{{∂}^{2}g}{∂s∂t}\right)}^2$

role="math" localid="1649938947871" $$\begin{gathered} det\left(H_g\right)=\left(2t{e}^{-s{t}^2}-4s{t}^3{e}^{-s{t}^2}+s^2{t}^5{e}^{-s{t}^2}\right)\left(-6s^{3}t{e}^{-s{t}^2}+4s^4{t}^3{e}^{-s{t}^2}\right)-{\left(2s{e}^{-s{t}^2}-2s^2{t}^2{e}^{-s{t}^2}+2s^3{t}^4{e}^{-s{t}^2}\right)}^2 \\ =6t^6{s}^5{e}^{-s{t}^2}-25t^4{s}^4{e}^{-s{t}^2}+16t^2{s}^3{e}^{-2s{t}^2}-4s^2{e}^{-2s{t}^2} \end{gathered}$$