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I need to find max and min of $f(x,y)=x^3 + y^3 -3x -3y$ with the following restriction: $x + 2y = 3$.

I used the multiplier's Lagrange theorem and found $(1,1)$ is the minima of $f$. Apparently, the maxima is $(-13/7, 17/7)$ but I could not find it via Lagrange's theorem.

Here's what I did:

I put up the linear system:

$\nabla f(x,y) = \lambda \, \nabla g(x,y)$

$g(x,y) = 0$

then,

$(3x^2 -3, 3y^2 -3) = \lambda (1,2)$

$x + 2y -3 = 0$

Solving for $\lambda$, I got $\lambda = 0$, which gave me $x = 1$ and $y = 1$.

How can I find the maxima if lambda only gives one value which is $0$?

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    $\begingroup$ no global maximum or global minimum. One local for each. As a one variable problem, take $x= 1 + 2 t,$ $ y = 1 - t$ as a parametrization of the constraint (line), calculate $x^3 + y^3 - 3x - 3y$ in terms of $t$ $\endgroup$
    – Will Jagy
    Dec 2 '20 at 17:45
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    $\begingroup$ "But I need to use Lagrange's theorem...". If I had known that, then I wouldn't have answered. Unfortunately, I'm not familiar with this theorem. I have therefore removed my answer. It would help if (for next time), you try to make your query very clear. $\endgroup$ Dec 2 '20 at 18:07
  • $\begingroup$ @user2661923 The last line of the question is: "How can I find the maxima if $\lambda$ only gives one value which is $0$?" It is clear that the OP is trying to solve this problem a certain way and got stuck. $\endgroup$
    – Théophile
    Dec 2 '20 at 18:26
  • $\begingroup$ @Théophile Yes, that was clear. What was not clear was that he was required to use the method for which he was stuck. Given how off-the-wall many mathSE queries are, re an OP often taking unusual avenues, I was fully justified in answering the query that he originally posed, which is "how do I determine the minimum and maximum values?" $\endgroup$ Dec 2 '20 at 18:34
  • $\begingroup$ @user2661923 Yes, there are many questions on MSE that are missing context or necessary information. This is not one of them; the OP specifically asked what went wrong with $\lambda$. You chose to take a different avenue because you are unfamiliar with Lagrange's Theorem. Rather than getting upset, why not take this as an opportunity to learn the theorem? :) $\endgroup$
    – Théophile
    Dec 2 '20 at 18:43
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As others have said, you don't necessarily need to use Lagrange multipliers. But since you've set up the system, we can see what happens:

$$ 3x^2-3=\lambda\\ 3y^2-3=2\lambda\\ x+2y-3=0 $$

From the first two equations, we have $3y^2-3=2(3x^2-3)$, which simplifies to $y^2-1=2(x^2-1)$. Rearranging the linear constraint, we have $x=3-2y$. Putting this information together leads to $$7y^2-24y+17=0.$$

You could solve this using the quadratic formula, but it is quicker to observe that $-24 = -7-17$:

$$7y^2-7y-17y+17=0$$

and so $(7y-17)(y-1)=0$.

This will give you the two local extrema.

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  • $\begingroup$ Right, I found the maxima as (-13/7, 17/7) but if I plug this pair into the hessian function (det = | (6x, 0), (0,6y)| and I get a negative value, which means this point is not local extrema. Why this contradiction? $\endgroup$ Dec 2 '20 at 18:26
  • $\begingroup$ @Brasilian_student Good question. Keep in mind that the point in question is not actually a critical point of $f(x,y)$ because it does not satisfy $f_x(x,y)=f_y(x,y)=0$. Rather, it is only an extreme point with respect to the restriction $x+2y=3$. On the other hand, $(x,y)=(1,1)$ is an extreme point of $f$. $\endgroup$
    – Théophile
    Dec 2 '20 at 18:39
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This is from your working -

$(3x^2 -3, 3y^2 -3) = \lambda (1,2)$

$3x^2 - 3 = \lambda, 3y^2-3 = 2\lambda$

Equating $\lambda$ from both equations,

$6x^2-6 = 3y^2-3 \implies 2x^2 - y^2 = 1$

Substitute $x$ from $x+2y = 3$

$2(3-2y)^2 - y^2 = 1$

$\implies 7y^2 - 24y + 17 = 0 \, $ or $(7y-17)(y-1) = 0$

Can you take it from here and find possible points for extrema?

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In this case, it is really not a great idea to use Lagrange multipliers. We can write $x$ in terms of $y$ (or vice-versa) using the restriction and reduce this question to a one variable optimisation problem. Substituting $x = 3 - 2y$, the problem reduces to finding the extrema of $f(y)=-7 y^3+36 y^2-51 y+18$. The critical points are $y = \frac{17}{7}$ and $y = 1$ and, considering that $f''(1)>0$ and $f''(\frac{17}{7})<0$, they are a local minimum and maximum, respectively.

It is also worth noticing that, since $\displaystyle \lim_{y \to \pm \infty} f(y) = \mp \infty$, there will not be any global extrema.

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    $\begingroup$ It is useful to leave some comment together with a downvote. I totally accept the down vote, but would be happy to improve the answer. $\endgroup$ Dec 2 '20 at 17:57
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    $\begingroup$ @AnindyaPrithvi Thanks for the feedback. I'll make some changes to the answer. Regarding the Lagrange multipliers, it was a very conscious option not to use them: I think that this a bad illustration of their use because the results are weaker than changing to a 1d problem. $\endgroup$ Dec 2 '20 at 22:12

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