MATH 251 Lecture 17

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Second Derivative Test

The second derivative depends partially on the Taylor series of the function. Given 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)} ,

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 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}

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 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) $\lambda >0,\mu <0$ or $\lambda <0,\mu >0$

Inconclusive (if determinant of matrix is 0)

1. cylinder: $\lambda \neq 0,\mu =0$ or $\lambda =0,\mu \neq 0$
2. 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 $f(x,y)=2x^{3}+xy^{4}+5x^{2}+y^{2}+4$

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