Question

A hessiana da função f dada por f (x,y) = 5x²y ² é

Answer 1

$H(x,y)= \left[\begin{array}{cc}f_{xx}&f_{xy}\\f_{yx}&f_{yy}\end{array}\right] \\\\\\f(x,y)=5x^2y^2\\\\f_x=10xy^2\\\\f_{xx}=10y^2\\\\f_{y}=10x^2y\\\\f_{yy}=10x^2\\\\f_{xy}=f_{yx}=20xy\\\\\\H(x,y)= \left[\begin{array}{cc}10y^2&20xy\\20xy&10x^2\end{array}\right] \\\\\\|H(x,y)|=100x^2y^2-400x^2y^2=-300x^2y^2$

Answer 2

✅ Tendo finalizado os devidos cálculos, concluímos que a matriz hessiana e o hessiano da referida função, são, respectivamente:

        $\Large\displaystyle\text{ \begin{gathered}\boxed{\boxed{\:\:\:\bf \mathcal{H}f(x, y) = \begin{bmatrix} 10y^{2}\:\:&\:\:20xy\\20xy\:\:&\:\:10x^{2}\end{bmatrix} \:\:\:}}\end{gathered} }$

             $\Large\displaystyle\text{ \begin{gathered}\boxed{\boxed{\:\:\:\bf Hf(x, y) = -300x^{2}y^{2}\:\:\:}}\end{gathered} }$

Seja a função em duas variáveis:

                            $\Large\displaystyle\begin{gathered} f(x, y) = 5x^{2}y^{2}\end{gathered}$

Para montarmos a matriz hessiana de uma função em duas variáveis devemos utilizar a seguinte fórmula

$\Large\displaystyle\begin{gathered} \bf I\end{gathered}$         $\Large\displaystyle\begin{gathered} \mathcal{H}f(x, y) = \begin{bmatrix} f_{xx}(x, y)\:\:&\:\:f_{xy}(x, y)\\f_{yx}(x, y)\:\:&\:\:f_{yy}(x, y)\end{bmatrix}\end{gathered}$

Para utilizar a fórmula dada devemos:

• Calcular a derivada segunda da função em termos de "x".

          $\Large\displaystyle\begin{gathered} f_{xx}(x, y) = f_{x}\left[f_{x}(x, y)\right]\end{gathered}$

                              $\Large\displaystyle\begin{gathered} = f_{x}\left[2\cdot5\cdot x^{2 - 1}\cdot y^{2}\right]\end{gathered}$

                             $\Large\displaystyle\begin{gathered} = f_{x}\left[10xy^{2}\right]\end{gathered}$

                             $\Large\displaystyle\begin{gathered} = 1\cdot10\cdot x^{1 - 1}\cdot y^{2}\end{gathered}$

                             $\Large\displaystyle\begin{gathered} = 10y^{2}\end{gathered}$

        $\Large\displaystyle\begin{gathered} \therefore\:\:f_{xx}(x, y) = 10y^{2}\end{gathered}$

• Calcular a derivada segunda da função em termos de "y".

            $\Large\displaystyle\begin{gathered} f_{yy}(x, y) = f_{y}\left[f_{y}(x, y)\right]\end{gathered}$

                                $\Large\displaystyle\begin{gathered} = f_{y}\left[2\cdot5\cdot x^{2}\cdot y^{2 - 1}\right]\end{gathered}$

                                $\Large\displaystyle\begin{gathered} = f_{y}\left[10x^{2}y\right]\end{gathered}$

                                $\Large\displaystyle\begin{gathered} = 1\cdot10\cdot x^{2}\cdot y^{1 - 1}\end{gathered}$

                                 $\Large\displaystyle\begin{gathered} = 10x^{2}\end{gathered}$

        $\Large\displaystyle\begin{gathered} \therefore\:\:f_{yy}(x, y) = 10x^{2}\end{gathered}$

• Calcular a derivada mista da função de "x" em termos de "y".

            $\Large\displaystyle\begin{gathered} f_{xy}(x, y) = f_{x}\left[f_{y}(x, y)\right]\end{gathered}$

                                $\Large\displaystyle\begin{gathered} = f_{x}\left[2\cdot5\cdot x^{2}\cdot y^{2 - 1}\right]\end{gathered}$

                                $\Large\displaystyle\begin{gathered} = f_{x}\left[10x^{2}y\right]\end{gathered}$

                                $\Large\displaystyle\begin{gathered} = 2\cdot10\cdot x^{2 - 1}\cdot y\end{gathered}$

                                 $\Large\displaystyle\begin{gathered} = 20xy\end{gathered}$

        $\Large\displaystyle\begin{gathered} \therefore\:\:f_{xy}(x, y) = 20xy\end{gathered}$

• Calcular a derivada mista da função de "y" em termos de "x".

          $\Large\displaystyle\begin{gathered} f_{yx}(x, y) = f_{y}\left[f_{x}(x, y)\right]\end{gathered}$

                              $\Large\displaystyle\begin{gathered} = f_{y}\left[2\cdot5\cdot x^{2 - 1}\cdot y^{2}\right]\end{gathered}$

                             $\Large\displaystyle\begin{gathered} = f_{y}\left[10xy^{2}\right]\end{gathered}$

                             $\Large\displaystyle\begin{gathered} = 2\cdot10\cdot x\cdot y^{2 - 1}\end{gathered}$

                             $\Large\displaystyle\begin{gathered} = 20xy\end{gathered}$

        $\Large\displaystyle\begin{gathered} \therefore\:\:f_{yx}(x, y) = 20xy\end{gathered}$

• Montar a matriz Hessiana. Para isso, devemos inserir na fórmula "I" os valores das derivadas segundas e das derivadas mistas. Então, temos:

             $\Large\displaystyle\begin{gathered} \mathcal{H}f(x, y) = \begin{bmatrix} 10y^{2}\:\:&\:\:20xy\\20xy\:\:&\:\:10x^{2}\end{bmatrix}\end{gathered}$

• Calcular o Hessiano "H" - determinante da matriz hessiana da função. Para isso, fazemos:

              $\Large\displaystyle\begin{gathered} Hf(x, y) = \det\mathcal{H}f(x, y)\end{gathered}$

                                   $\Large\displaystyle\begin{gathered} = 10y^{2}\cdot10x^{2} - 20xy\cdot20xy\end{gathered}$

                                   $\Large\displaystyle\begin{gathered} = 100x^{2}y^{2} - 400x^{2}y^{2}\end{gathered}$

                                   $\Large\displaystyle\begin{gathered} = -300x^{2}y^{2}\end{gathered}$

           $\Large\displaystyle\begin{gathered} \therefore\:\:Hf(x, y) = -300x^{2}y^{2}\end{gathered}$

 

$\LARGE\displaystyle\text{ \begin{gathered} \underline{\boxed{\boldsymbol{\:\:\:Bons \:estudos!!\:\:\:Boa\: sorte!!\:\:\:}}}\end{gathered} }$

Saiba mais sobre matriz hessiana:

1. https://brainly.com.br/tarefa/16334308