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`.