Question

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

Answer 1

$f(x,y)=2y^3+12x^2-24xy$
Find the first derivatives:
$f'_x=24x-24y \\ f'_y=6y^2-24x$.
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 $A_1(0,0), \\ A_2(4,4)$.

Find the first derivatives:
$f^{''}_{xx}=24 \\ f^{''}_{xy}=f^{''}_{yx}=-24 \\ f^{''}_{yy}=12y$ 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 $f(4,4)=-64$.