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	FriCAS Library Examples
ExampleCompositeFunction ←	↑	→ ExampleInequalitiesViaCAD

ExampleExtrema

Edit detail for ExampleExtrema revision 1 of 1

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Editor: test1 Time: 2024/11/18 13:07:09 GMT+0
Note:

changed:
-
We would like to find extrama of  $x^2 + y^2 − 2x + 4y + 5$ on the curve with equation
$2x^4 + 3y^7 = 4$.  We set up and solve system of equations for Lagrange's multipliers:
\begin{axiom}
f := x^2 + y^2 - 2*x + 4*y + 5
g := 2*x^4 + 3*y^7 - 4
dfx := D(f, x) - l*D(g,x)
dfy := D(f, y) - l*D(g, y)
)set output tex off
)set output algebra on
r1 := solve([g, dfx, dfy], 1.0e-15)
\end{axiom}

Now we substitute values info $f$:
\begin{axiom}
eval(f, r1(1))
eval(f, r1(2))
eval(f, r1(3))
\end{axiom}

One can see that $f$ goes to infinity when $x$ and $y$ go to infinity, so the third point gives minimal value,
that is $f$ attains minimal value $1.0407113140\dots$ at $x = 1.5513349690\dots$, $y=- 1.1416637803\dots$.
Consequently the second point gives local minimum and first gives local maximum.

We would like to find extrama of on the curve with equation . We set up and solve system of equations for Lagrange's multipliers:

fricas

(1) -> f := x^2 + y^2 - 2*x + 4*y + 5

$\label{eq1}{{y}^{2}}+{4 \ y}+{{x}^{2}}-{2 \ x}+ 5$

(1)

Type: Polynomial(Integer)

fricas

g := 2*x^4 + 3*y^7 - 4

$\label{eq2}{3 \ {{y}^{7}}}+{2 \ {{x}^{4}}}- 4$

(2)

Type: Polynomial(Integer)

fricas

dfx := D(f, x) - l*D(g,x)

$\label{eq3}-{8 \ l \ {{x}^{3}}}+{2 \ x}- 2$

(3)

Type: Polynomial(Integer)

fricas

dfy := D(f, y) - l*D(g, y)

$\label{eq4}-{{21}\ l \ {{y}^{6}}}+{2 \ y}+ 4$

(4)

Type: Polynomial(Integer)

fricas

)set output tex off

fricas

)set output algebra on

r1 := solve([g, dfx, dfy], 1.0e-15)

   (5)
   [
     [
         y
       = 
        0.9208297689_3710153170_2034169691_3819039444_1382982572_8057479579_008
        7622506_6610299122_7720402279_4236135414_3512919130_1925525120_59548528
        21_0249360484
       ,
      x = - 1.0373393332_54388858,

         l
       = 
        0.4562898854_2744960049_4865614063_3636455024_1672207281_8815376923_522
        6096440_0032082177_9819858606_1445957048_6060348282_1979071104_57890667
        02_127456665
       ]
     ,

     [y = - 0.8526080934_9016125576, x = - 1.2563366987_746125493,

         l
       = 
        0.2844629945_1171337269_5579861614_8489100133_6205527230_4243593768_166
        3989631_1268805817_6205268127_7290243939_8182115881_6636341271_07798121
        86_9564056396
       ]
     ,

     [y = - 1.1416637803_267151789,

         x
       = 
        1.5513349690_2801873789_3770322428_2563302905_0169770846_7275566242_580
        5498910_8857124756_6803091958_3494850823_0917955654_4637921892_90514021
        58_240811401
       ,

         l
       = 
        0.0369180540_4167569804_0713896075_2132995192_5078820127_9289458299_138
        6283076_5542713778_8059588147_7007596438_6175822184_9989070051_37880332
        76_7963409423_8
       ]
     ]

Type: List(List(Equation(Polynomial(Float))))

Now we substitute values info :

fricas

eval(f, r1(1))

   (6)  12.6819980979_34599666

Type: Polynomial(Float)

fricas

eval(f, r1(2))

   (7)  6.4075634853_613991814

Type: Polynomial(Float)

fricas

eval(f, r1(3))

   (8)  1.0407113140_76151838

Type: Polynomial(Float)

One can see that goes to infinity when and go to infinity, so the third point gives minimal value, that is attains minimal value $1.0407113140\dots$ at $x = 1.5513349690\dots$ , $y=- 1.1416637803\dots$ . Consequently the second point gives local minimum and first gives local maximum.