MATH 251 Lecture 17

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Second Derivative Test

The second derivative depends partially on the Taylor series of the function. Given $f(x,y)$ ,

$T_{1}(x_{0}+h,y_{0}+k)=f(x_{0},y_{0})+{\frac {\partial f}{\partial x}}(x_{0},y_{0})\cdot h+{\frac {\partial f}{\partial y}}(x_{0},y_{0})\cdot k$

$T_{2}(x_{0}+h,y_{0}+k)=T_{1}(x,y)+{\frac {1}{2}}\left({\frac {\partial ^{2}f}{\partial x^{2}}}(x_{0},y_{0})\cdot h^{2}+2{\frac {\partial ^{2}f}{\partial x\partial y}}(x_{0},y_{0})\cdot hk+{\frac {\partial ^{2}f}{\partial y^{2}}}(x_{0},y_{0})\cdot k^{2}\right)$

The partial derivatives of the stuff in "()" are coefficients for variation in $x$ and $y$

From linear algebra, $ah^{2}+2bhk+ck^{2}$ is equal to the following matrix multiplication:

{\begin{pmatrix}h&k\end{pmatrix}}{\begin{pmatrix}a&b\\b&c\end{pmatrix}}{\begin{pmatrix}h\\k\end{pmatrix}}

For small variations in $h$ and $k$ , we find the directions in which $f(x,y)$ curves around critical points:

$Q(h,k)={\frac {1}{2}}\left(\lambda h^{2}+\mu k^{2}\right)$ , where $\lambda =f_{xx}$ and $\mu =f_{yy}$ , assuming $f_{xy}=f_{yx}=0$

Consider the hessian matrix ${\begin{pmatrix}\lambda &0\\0&\mu \end{pmatrix}}$

Conclusive

Minimum: (determinant is positive) $\lambda ,\mu >0$
Maximum: (determinant is positive) $\lambda ,\mu <0$
Saddle Surface: (determinant is negative) $\lambda >0,\mu <0$ or $\lambda <0,\mu >0$

Inconclusive (if determinant of matrix is 0)

cylinder: $\lambda \neq 0,\mu =0$ or $\lambda =0,\mu \neq 0$
plane: $\lambda =\mu =0$

By the way, in almost all cases, $f_{xy}=f_{yx}$ .

Linear Algebra Stuff

If $f_{xy}\neq 0$ , then by a rotation of coordinates $x'=\cos \theta x+\sin \theta y,y'=\ldots$ , f_{x'y'} = 0

Example

Find and classify any Critical points of the function $f(x,y)=2x^{3}+xy^{4}+5x^{2}+y^{2}+4$

${\begin{aligned}f_{x}=6x^{2}+y^{4}+10x&=0\\f_{y}=2xy+2y&=0\\y=0{\mbox{ or }}x=-1\\{\mbox{if }}y=0,&\ x=0,-{\frac {5}{3}}\\{\mbox{if }}x=-1,&\ y=\pm 2\\\end{aligned}}$