Critical point of function of two variables

GreenGoblin

Member
Hello,
Thank you

$f(x,y) = 2cos(2x) + sin(x^{2}-y^{2})$

Find all the first and second order derivatives, hence show the origin is a critical point and find which type of critical point
First time attempting critical point question, I know it has to do with second derivative but I am not sure on what the definition is,

I made the derivatives (I don't have any way to verify this other that my own mind and it needs to be right to check the critical point criteria I figure, so please point out to me any mistake)

$df/dx = -4sin(2x) + 2xsin(x^{2})cos(y^{2}) - cos(x^{2})sin(y^{2})$

$df/dy = -2y(sin(y^{2})sin(x^{2}) + cos(y^{2})cos(x^{2}))$

$d^{2}f/dx^{2} = -8cos(2x) + 4x^{2}cos(x^{2})(sin(y^{2}) + cos(y^{2})) + 2sin(x^{2})(sin(y^{2}) + cos(y^{2}))$

dwsmith

Well-known member
Hello,
Thank you

$f(x,y) = 2cos(2x) + sin(x^{2}-y^{2})$

Find all the first and second order derivatives, hence show the origin is a critical point and find which type of critical point
First time attempting critical point question, I know it has to do with second derivative but I am not sure on what the definition is,

I made the derivatives (I don't have any way to verify this other that my own mind and it needs to be right to check the critical point criteria I figure, so please point out to me any mistake)

$df/dx = -4sin(2x) + 2xsin(x^{2})cos(y^{2}) - cos(x^{2})sin(y^{2})$

$df/dy = -2y(sin(y^{2})sin(x^{2}) + cos(y^{2})cos(x^{2}))$

$d^{2}f/dx^{2} = -8cos(2x) + 4x^{2}cos(x^{2})(sin(y^{2}) + cos(y^{2})) + 2sin(x^{2})(sin(y^{2}) + cos(y^{2}))$
http://en.wikipedia.org/wiki/Second_partial_derivative_test

check out the example.

GreenGoblin

Member
Hi,
Thanks,

Why is it that the second derivative of x is used to find the critical point type but not y? What property is it that makes x more relevant than y in this case? (since the function is of both variables I don't understand why x is more involved in the evaluation than y..?)

dwsmith

Well-known member
Hi,
Thanks,

Why is it that the second derivative of x is used to find the critical point type but not y? What property is it that makes x more relevant than y in this case? (since the function is of both variables I don't understand why x is more involved in the evaluation than y..?)
What you find with x, you plug into the y derivative. You can start with y as well.

GreenGoblin

Member
No no, what I mean is, why is it the $d^{2}f/dx^{2}$ (or $f_{xx}$ whatever notation you prefer) is used to find out whether its a maximum or minimum? I can't see that x or y should be any different since theyre both independent variables? But it specificies the 2nd x derivative is to be used. What makes this the case?

dwsmith

Well-known member
No no, what I mean is, why is it the $d^{2}f/dx^{2}$ (or $f_{xx}$ whatever notation you prefer) is used to find out whether its a maximum or minimum? I can't see that x or y should be any different since theyre both independent variables? But it specificies the 2nd x derivative is to be used. What makes this the case?
That is the determinant for the Hessian matrix.
For example, the det of
$$\begin{vmatrix}a & b \\ c & d \end{vmatrix} = ad-bc$$

Examine at f_xx has to with positive and negative definiteness.

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GreenGoblin

Member
No, the determinant is used for the first step. I'm asking about the second. Given it is a maximum OR minimum, and not a saddle (from using the determinant), the process for determining which of maximum or minimum it is. I'm querying why the second x derivative is specified as being used and not y (since x and y are in essence, the same for problems like this. You could switch them around with no effect, so why isn't this true for that as well).

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Jester

Well-known member
MHB Math Helper
To answer your question, yes you can use either $f_{xx}$ or $f_{yy}$. It doesn't matter.

GreenGoblin

Member
To answer your question, yes you can use either $f_{xx}$ or $f_{yy}$. It doesn't matter.
Thank you

I like to have a justification for a formula rather than just a formula itself, I couldn't tally this as written. I saw no mention of this in the source provided.

Gracias,
GreenGoblin

HallsofIvy

Well-known member
MHB Math Helper
The condition for max or min is that $f_{xx}f_{yy}- f_{xy}^2$ be greater than 0. Since $f_{xy}$ is squared, it is non-negative so that $-f_{xy}^2$ is non-positive. In order that
$f_{xx}f_{yy}- f_{xy}^2$ be greater than 0, then, $f_{xx}f_{yy}$ must be positive, which, in turn means that $f_{xx}$ and $f_{yy}$ must have the same sign- so it is sufficient to check either of them to see whether it is a max or min.

The problem with a justification for this formula (which typically just "given" in a calculus text without proof or justification) is that it requires some pretty deep linear algebra. In a strict sense, "the" derivative, at a given point, of a function of two variables (as opposed to the partial derivatives) is a linear transformation from $R^2$ to R which can be represented by the vector $\begin{bmatrix}\frac{\partial f}{\partial x} & \frac{\partial f}{\partial y}\end{bmatrix}$, the "gradient" of the function f.

And, then, the second derivative is the linear transformation from $R^2$ to $R^2$ which can be represented by the matrix
$\begin{bmatrix}\frac{\partial^2f}{\partial x^2} & \frac{\partial^2}{\partial x\partial y} \\ \frac{\partial^2f}{\partial x\partial y} & \frac{\partial^2f}{\partial y^2}\end{bmatrix}$

Now, that is a symmetric matrix so there exist a basis (coordinate system) in which it becomes the diagonal matrix
$\begin{bmatrix}\frac{\partial^2f}{\partial x'^2} & 0 \\ 0 & \frac{\partial^2f}{\partial y^2}\end{bmatrix}$

Further, the determinants of those two matrices are equal. Now that means that $f_{xx}f_{yy}- f_{xy}^2= f_{x'x'}f_{y'y'}$ which in turn means that $f_{fxx}f_{yy}- f_{xy}^2$ will be positive if and only if there exist a coodinate system, x'y', such that $f_{x'x'}$ and $f_{y'y'}$ have the same sign. Of course, you are only check this if the first derivatives, $f_x$ and $f_y$ are 0. So, in terms of the Taylor's series, $f(x',y')= f(x'_0, y'_0)+ f_{x'x'}(x'- x'_0)^2+ f_{y'y'}(y'- y'_0)^2$ plus higher powers of x' and y'. For x' and y' sufficiently close to $x'_0$ and $y'_0$ those higher power terms are negligible. And, if $f_{x'x'}$ and $f_{y'y'}$ are both positive, $f(x'_0, y'_0)+ f_{x'x'}(x'- x'_0)^2+ f_{y'y'}(y'- y'_0)^2$ is a parabola opening upward, so we have a minimum at $(x'_0, y'_0)$ while if they are negative we have a parabola opening downward so we have a maximum.

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