Question

Find the stationary points of
f(x, y) = 5x2 + 10y2 + 12xy - 4x - 6y +1.​

Answer 1

Answer:

I don't have what you want take this as example

Answer 2

Answer:

Here we will find the stationary points for the given function $f(x, y) = 5x2 + 10y2 + 12xy - 4x - 6y +1$

Stationary Point is $(\frac{1}{7},\frac{3}{14} )$

Step-by-step explanation:

To find the stationary points we should do the following steps.

Step 1:first find $f'(x)$. which means that differentiate the given equation based on $x$.

Step 2:next find $f'(y)$. which means that differentiate the given equation based on $y$.

Step 3:equate $f'(x)=0$ and $f'(y)=0$ and solve the equations.

Step 4:then we can find the stationary points

Step 1:first find $f'(x)$:

Differentiate the given equation based on $x$. That is,

           $f'(x)=10x+12y-4$

Step 2:next find $f'(y)$:

Differentiate the given equation based on $y$. That is,

           $f'(y)=20y+12x-6$

Step 3:equate $f'(x)=0$ and $f'(y)=0$ and solve this two equations.

          $f'(x)=0$ ⇒    $10x+12y-4=0$. It can be written as,

                                       $10x+12y=4$

the above equation is divided by $2$. So that,

                                           $5x+6y=2$   ----------(1)

          $f'(y)=0$ ⇒    $20y+12x-6=0$ .It can be written as,

                                       $20y+12x=6$ .interchange $x,y$ terms

                                        $12x+20y=6$  

the above equation is divided by $2$. So that,

                                          $6x+10y=3$  -----------(2)

equation($1$) multiply with $6$. It can be written as,

                                       $30x+36y=12$ -----------(3)

equation($2$) multiply with $5$. It can be written as,

                                       $30x+50y=15$ -----------(4)

Subtract equation($4$) - equation($3$)

                                                 $14y=3$

                                                    $y=\frac{3}{14}$

Apply the value of $y$ in equation($1$). Then we can get the value of $x$.

            equation($1$) ⇒  $5x+6(\frac{3}{14} )=2$

                                      $5x+3(\frac{3}{7} )=2$

                                        $5x+(\frac{9}{7} )=2$

                                                  $5x=2-(\frac{9}{7} )$

                                                  $5x=\frac{14-9}{7}$

                                                  $5x=\frac{5}{7}$

                                                    $x=\frac{5}{5*7}$

                                                    $x=\frac{1}{7}$

Step 4:then we can find the stationary points

          Stationary Point is $(\frac{1}{7},\frac{3}{14} )$