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Re: Local extrema of a function of two variables
*To*: mathgroup at smc.vnet.net
*Subject*: [mg90017] Re: Local extrema of a function of two variables
*From*: "Jean-Marc Gulliet" <jeanmarc.gulliet at gmail.com>
*Date*: Thu, 26 Jun 2008 04:44:49 -0400 (EDT)
On Wed, Jun 25, 2008 at 4:56 PM, Barun von Doppeldecker
<nikoi8888 at gmail.com> wrote:
> if you still want to help. here is another problem i came across today: i
> have a function f(x, y) = 4xy â?? x^4 â?? y^4 and i have to do three things.
> First i have to find candidates for local extremes, then i have to exactly
> pinpoint that extremes and finally plot a graf surrounding that point of
> extremum.
> I have found candidates, but the other two things i just can't....Here is
> what i've done:
>
> f[x_,y_]:=4 x y-x^4-y^4
>
> Solve[{D[f[x,y],x]Å 0,D[f[x,y],y]Å 0},{x,y}]
>
> if you could help me with other 2 things, i would be really grateful... if
> not, thank you anyway.... ;)
<snip>
[... Reply cross-posted to MathGroup ...]
You could follow the following "template".
(* Say that the function we want to study is the following: *)
f[x_, y_] := 4 x y - x^4 - y^4
(* We compute the first derivatives of f w.r.t.to x and y: *)
gradient = D[f[x, y], {{x, y}, 1}]
{-4 x^3 + 4 y, 4 x - 4 y^3}
(* We equate each derivative to zero and solve for x and y: *)
sols = Solve[gradient == 0]
{{x -> -1, y -> -1}, {x -> 0, y -> 0}, {x -> -I,
y -> I}, {x -> I, y -> -I}, {x -> 1, y -> 1}, {x -> -(-1)^(1/4),
y -> -(-1)^(3/4)}, {x -> (-1)^(1/4),
y -> (-1)^(3/4)}, {x -> -(-1)^(3/4),
y -> -(-1)^(1/4)}, {x -> (-1)^(3/4), y -> (-1)^(1/4)}}
(* We want only the real points,so we discard the complex \
solutions: *)
pts =
Cases[sols, {x -> vx_, y -> vy_} /; Im[vx] == 0 && Im[vy] == 0]
{{x -> -1, y -> -1}, {x -> 0, y -> 0}, {x -> 1, y -> 1}}
(* So we have the critical points.
Now,we compute the Hessian matrix, *)
hessian = D[f[x, y], {{x, y}, 2}]
{{-12 x^2, 4}, {4, -12 y^2}}
(* and take its determinant: *)
d = hessian // Det
-16 + 144 x^2 y^2
(* Then we compute the values of the Hessian determinant at \
the critical points: *)
d /. pts
{128, -16, 128}
D[f[x, y], {x, 2}] /. pts
{-12, 0, -12}
(* We apply the second derivative test.
Since the Hessian is negative at the point (0,0),we \
conclude that this is a saddle point.
Since the Hessian is positive at \
the points (-1, -1) ant (1,1),and the second derivatives w.r.t. x at
these points are negative, we conclude that those points are local maxima. *)
(* We can "check" (or anticipate) these results by \
plotting the function: *)
Plot3D[f[x, y], {x, -2, 2}, {y, -2, 2}, PlotRange -> All]
ContourPlot[f[x, y], {x, -2, 2}, {y, -2, 2},
ColorFunction -> "SunsetColors"]
[keywords: function two variables local minimum maximum critical point
Hessian calculus]
Regards,
--
Jean-Marc
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