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I am trying to find the critical points of the function:

$f(x,y)=2x^4-3x^2y+y^2$

and find the Max, Min and saddle points. What I've done so far is:

$f_x=8x^3-6xy=0 , f_y=-3x^2+2y=0 , f_{xx}=24x^2-6y , f_{yy}=2 , f_{xy}=-6x$

So (0,0) is the only critical point. But using the second partial derivative test:

$\Delta(0,0)=f_{xx} . f_{yy} - f^2_{xy}=0$

Which is inconclusive. Without using MATLAB or similar software and based on calculation, how can we determine whether (0,0) is Max, Min or saddle point?

Or the general question is what to do when it's inconclusive (without using software)?

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4 Answers 4

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Plug in $x = \epsilon$, $y = 3\epsilon^2$ and you will get that the function is greater than $0$ for all $\epsilon > 0$. Plug in $x = (3/4) \epsilon$, $y = \epsilon ^2$ and you will get that the function is less than $0$ for all $\epsilon > 0$. This implies that $(0,0)$ must be a saddle point because you take take $\epsilon > 0$ arbitrarily small.

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  • $\begingroup$ If $x=\dfrac{2\epsilon}{3}$ and $y=\epsilon^2$, then $f(x,y)=\dfrac{5\epsilon^4}{81}$, which is positive, not negative. $\endgroup$
    – Steve Kass
    Apr 13, 2014 at 21:43
  • $\begingroup$ Your suggestion works if you change $\frac{2}{3}$ to $\frac{3}{4}$. For any $\epsilon>0$, $\,f(\dfrac{3\epsilon}{4},\epsilon^2)=-\dfrac{7\epsilon^4}{128}$. $\endgroup$
    – Steve Kass
    Apr 13, 2014 at 21:59
  • $\begingroup$ How did you calculate y if $x=\dfrac{3\epsilon}{4}$? Shouldn't we use the equation $y=\dfrac{3x^2}{2}$? @SteveKass $\endgroup$
    – Sam
    Apr 13, 2014 at 22:06
  • $\begingroup$ I didn’t calculate $y$. I calculated $f(x,y)=2x^4-3x^2y+y^2$ at the point $(x,y)=(\dfrac{3\epsilon}{4},\epsilon^2)$. $\endgroup$
    – Steve Kass
    Apr 13, 2014 at 22:08
  • $\begingroup$ So you chose an arbitrary value ($\varepsilon^2$) for y? (I can't figure out how you got $y=\varepsilon^2$) @SteveKass $\endgroup$
    – Sam
    Apr 13, 2014 at 22:11
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Hint: $$2x^4−3x^2y+y^2=(x^2-y)(2x^2-y).$$

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Possibly easier hint (complete the square): $$ 2x^4 - 3x^2 + y^2 = (y-\frac32 x^2)^2 - \frac14x^2 $$

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  • $\begingroup$ Still no clue, would you please elaborate? @mrf $\endgroup$
    – Sam
    Apr 13, 2014 at 20:55
  • $\begingroup$ Same idea that my answer: the function can take $>0$ and $<0$ values for $(x,y)$ arbitrarily near of $(0,0)$. $\endgroup$
    – Martín-Blas Pérez Pinilla
    Apr 13, 2014 at 21:03
  • $\begingroup$ I see, so can we say if the test is inconclusive, we have to examine the values in a neighborhood of the critical point? @mrf $\endgroup$
    – Sam
    Apr 13, 2014 at 21:06
  • $\begingroup$ @Sam, usually is the best idea. $\endgroup$
    – Martín-Blas Pérez Pinilla
    Apr 14, 2014 at 6:44
  • $\begingroup$ @mrf..can you please explain me how you complete the square?It is not like quadratic.I am in mess finding how to do in two variable case. $\endgroup$
    – Believer
    Dec 29, 2017 at 20:01
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Added: Here's a way not to solve the problem (because it gives the wrong answer!)

Look at the concavity of the function at $(0,0)$ in a non-axis direction $y=ax$. $$\left.\frac{d^2}{dx^2}\right\rvert_{x=0}\left(f(x,ax)\right) = \left.\frac{d^2}{dx^2}\right\rvert_{x=0}\left(2x^4-3x^2 (a x)+(a x)^2\right)=\left. 2a^2-18ax+24x^2\right\rvert_{x=0}=2a^2.$$ This tells you that the function is concave up at $(0,0)$ in every non-axis direction, so $(0,0)$ is not a saddle point. It's not a local maximum, because $f_{yy}(0,0)>0$, so it’s a local minimum.

(The right answer is that this is a saddle point. See the answer @user2566092 gave, and the correction in the comments, if the answer hasn't been fixed.)

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  • $\begingroup$ Wait sorry why doesn't this method work? Is it because you've only considered the concavity of the function in a non-axis linear direction, (so it looks concave up in every linear direction) however its like concave down along some non-linear curves? $\endgroup$
    – QCD_IS_GOOD
    May 8, 2016 at 11:07
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    $\begingroup$ Sort of. I just wanted to point out that up-concavity in every linear direction through a critical point is not sufficient evidence for a local minimum. You cannot conclude that you have a local minimum $f(0,0)=0$ just because along every linear direction through $(0,0)$, the function is either constant or concave up — that’s not a theorem. (This function is a counterexample to show that that conclusion is not valid in general.) You have here a function that is never concave down in a linear direction through $(0,0)$ but nevertheless has points arbitrarily close to zero where $f(x,y) < 0$. $\endgroup$
    – Steve Kass
    May 9, 2016 at 1:37

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