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

1. Minimum: (determinant is positive) $\lambda ,\mu >0$
2. Maximum: (determinant is positive) $\lambda ,\mu <0$
3. Saddle Surface: (determinant is negative) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda > 0, \mu < 0} or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda < 0, \mu > 0}

Inconclusive (if determinant of matrix is 0)

1. cylinder: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda \ne 0, \mu = 0} or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda = 0, \mu \ne 0}
2. plane: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda = \mu = 0}

By the way, in almost all cases, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_{xy} = f_{yx}} .

Linear Algebra Stuff

If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_{xy} \ne 0} , then by a rotation of coordinates Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x' = \cos\theta x + \sin\theta y, y' = \ldots} , f_{x'y'} = 0

Example

Find and classify any Critical points of the function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x,y) = 2x^3+ xy^4+5x^2+y^2+4}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} 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{align}}

So the critical points are (-5/3, 0), (0, 0), (-1, 2), (-1, -2)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} f_{xx} = 12x+10 \\ f_{xy} = 2y \\ f_{yy} = 2x+2 \end{align}}

So the Hessian matrix is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{pmatrix}12x+10&2y\\2y&2x+2\end{pmatrix}} , and the determinant at critical points are:

1. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 10/3 > 0} (min/max); (max since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_{xx}(-\tfrac{5}{3},0) < 0} )
2. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 20 > 0} (min/max); (min since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_{xx}(0,0) > 0} )
3. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -16 < 0} (Saddle)
4. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -16 < 0} (Saddle)

The values at each of the critical points are:

1. 233/27
2. 4
3. 7
4. 7