Please help me i dont understand this is 15 points

Mathematics

1 answer:

matrenka [14]3 years ago

3 0

Answer:

OPTION 2 IS THE CORRECT ANSWER

Step-by-step explanation:

ITS SIMPLE HOPE U GET IT

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Find all local extrema for f(x, y) = 2y3 + 12x2 − 24xy. (if an answer does not exist, enter dne.) local minimum (x, y) = local m

OlgaM077 [116]

$f(x,y)=2y^3+12x^2-24xy$
Find the first derivatives:
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Solve the system $\left \{ {{f'_x=0} \atop {f'_y=0}} \right.$ :
$\left \{ {{24x-24y=0} \atop {6y^2-24x=0}} \right. \rightarrow \left \{ {{x=y} \atop {6y^2-24y=0}}$ . The second equation has solutions $y_1=0, \\ y_2=4$ and then $x_1=0, \\ x_2=4$ and you have two points $A_1(0,0), \\ A_2(4,4)$ .

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$f^{''}_{xx}=24 \\ f^{''}_{xy}=f^{''}_{yx}=-24 \\ f^{''}_{yy}=12y$ and calculate $\Delta=\left| \left[\begin{array}{cc}24&-24\\-24&12y\end{array}\right]\right |=24\cdot 12y-(-24)^2=288y-576$ .
Since $\Delta(A_1)=-576$ and $f^{''}_{xx}\ \textgreater \ 0$ , $A_1$ is a point of maximum and $f(0,0)=0$ .
Since $\Delta(A_2)=576$ and $f^{''}_{xx}\ \textgreater \ 0$ , $A_1$ is a point of minimum and $f(4,4)=-64$ .

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Step-by-step explanation: