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 July 15th, 2017, 12:02 PM #1 Senior Member     Joined: Nov 2015 From: United States of America Posts: 165 Thanks: 21 Math Focus: Calculus and Physics Second derivative test conclusions Hello forum, In multivariable calculus, when finding critical points and utilizing the second derivative test, if you are trying to determine if a critical point is a max, min, or saddle, you find D = D(a,b) = $\dfrac{\partial^2 f}{\partial x^2}$ $\dfrac{\partial^2 f}{\partial y^2}$ - $(\dfrac{\partial^2 f}{\partial xy})^2$ If D is any real value between negative and positive infinity, but $\dfrac{\partial^2 f}{\partial x^2}$ = 0, what could I conclude about the second derivative test here? My book only gives me cases for when $\dfrac{\partial^2 f}{\partial x^2}$ > 0 or $\dfrac{\partial^2 f}{\partial x^2}$ < 0. Thank you for taking the time to read my post and help me out. Jacob
 July 15th, 2017, 12:29 PM #2 Senior Member   Joined: Oct 2009 Posts: 189 Thanks: 74 The test in inconclusive. Compare with a real valued function $f:\mathbb{R}\rightarrow \mathbb{R}$ with $f'(a) = f''(a) = 0$, then it is also inconclusive. For example, take $f(x) = x^3$, then we have a saddle at $0$. While if $f(x) = x^4$, then we have a maximum at $0$. Note that in both cases, $f'(0) = f''(0) = 0$. To conclude whether you have a saddle, minimum or maximum, you'll need to check higher derivatives. Thanks from SenatorArmstrong
 July 15th, 2017, 12:33 PM #3 Senior Member   Joined: Oct 2009 Posts: 189 Thanks: 74 Note that if $H:= f_{xx}f_{yy} - (f_{xy})^2 > 0$, then $f_{xx}$ has to be nonzero. If $f_{xx} = 0$, then $H = -(f_{xy})^2 < 0$. Thanks from SenatorArmstrong
July 15th, 2017, 02:12 PM   #4
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 Originally Posted by Micrm@ss The test in inconclusive. Compare with a real valued function $f:\mathbb{R}\rightarrow \mathbb{R}$ with $f'(a) = f''(a) = 0$, then it is also inconclusive. For example, take $f(x) = x^3$, then we have a saddle at $0$. While if $f(x) = x^4$, then we have a maximum at $0$. Note that in both cases, $f'(0) = f''(0) = 0$. To conclude whether you have a saddle, minimum or maximum, you'll need to check higher derivatives.
Thanks a lot for helping me out.

What could I conclude from say $\dfrac{\partial^3 f}{\partial x^3}$ or $\dfrac{\partial^3 f}{\partial y^3}$ ?

For example... $f(x,y) = xy^2 + x^2 y + 8xy$

Graphically, I see there is a critical point at $f(x,y) = f(0,0)$

Also algebraically, $\dfrac{\partial f}{\partial x} = y^2 + 2xy+ 8y$

By factoring the $y$ out, I noticed that $\dfrac{\partial f}{\partial x} =y^2 + 2xy+ 8y = 0$ , when $y=0$

The same can be said about $\dfrac{\partial f}{\partial y}$ except this time $x$ will be factored out.

$\dfrac{\partial f}{\partial y} = x^2 + 2xy + 8x = 0$ when $x=0$.

$\dfrac{\partial f^2}{\partial y^2} = 2x$ and $\dfrac{\partial f^2}{\partial x^2}=2y$

So when $f(x,y) = f(0,0)$ the second derivative test is inconclusive for those values. Additionally, $\dfrac{\partial^3 f}{\partial x^3}$ and $\dfrac{\partial^3 f}{\partial y^3} = 0$

What would be an advisable approach at this point in the problem to help drive the point home that $(0,0)$ is a critical point on $f(x,y)$. Unless it is not a critical point and I made a mistake?

Thanks a ton,

Jacob

July 15th, 2017, 02:29 PM   #5
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Quote:
 Originally Posted by SenatorArmstrong Thanks a lot for helping me out. What could I conclude from say $\dfrac{\partial^3 f}{\partial x^3}$ or $\dfrac{\partial^3 f}{\partial y^3}$ ? For example... $f(x,y) = xy^2 + x^2 y + 8xy$ Graphically, I see there is a critical point at $f(x,y) = f(0,0)$ Also algebraically, $\dfrac{\partial f}{\partial x} = y^2 + 2xy+ 8y$ By factoring the $y$ out, I noticed that $\dfrac{\partial f}{\partial x} =y^2 + 2xy+ 8y = 0$ , when $y=0$ The same can be said about $\dfrac{\partial f}{\partial y}$ except this time $x$ will be factored out. $\dfrac{\partial f}{\partial y} = x^2 + 2xy + 8x = 0$ when $x=0$. $\dfrac{\partial f^2}{\partial y^2} = 2x$ and $\dfrac{\partial f^2}{\partial x^2}=2y$ So when $f(x,y) = f(0,0)$ the second derivative test is inconclusive for those values. Additionally, $\dfrac{\partial^3 f}{\partial x^3}$ and $\dfrac{\partial^3 f}{\partial y^3} = 0$ What would be an advisable approach at this point in the problem to help drive the point home that $(0,0)$ is a critical point on $f(x,y)$. Unless it is not a critical point and I made a mistake? Thanks a ton, Jacob
But $f_{xy}\neq 0$, so $H = f_{xx} f_{yy} - (f_{xy})^2 < 0$. So the second derivative test works. It only doesn't work when $H=0$.

July 15th, 2017, 02:53 PM   #6
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 Originally Posted by Micrm@ss But $f_{xy}\neq 0$, so $H = f_{xx} f_{yy} - (f_{xy})^2 < 0$. So the second derivative test works. It only doesn't work when $H=0$.
Okay thank you. And since in my case $H<0$ it is not a local maximum or minimum. That is strange though, since wolfram does consider it a critical point.

July 15th, 2017, 02:54 PM   #7
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 Originally Posted by SenatorArmstrong Okay thank you. And since in my case $H<0$ it is not a local maximum or minimum. That is strange though, since wolfram does consider it a critical point.
It's a saddle point. That's counted as a critical point.

July 15th, 2017, 03:22 PM   #8
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 Originally Posted by Micrm@ss It's a saddle point. That's counted as a critical point.
Thanks for all the help

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