Problemata optimizationis

Svmmarivm

Haec lectio problemata tractat in qvibvs optima ex mvltis configvrationibvs possibilibvs invenienda est: rectangvlvm maximae areae, cylindrvs minimae svperficiei, via brevissima vel celerrima. Lectiones svperiores instrvmenta dedervnt ad extrema absolvta fvnctionis datae invenienda; hic fvnctio non datvr, et plvrimvm laboris, atqve errorvm, in ea constrvenda consistit. Lectio methodvm qvinqve articvlorvm statvit (variabilem, fvnctionem, dominivm, extrema absolvta cvm sva ivstificatione, conclvsionem) et demonstrat fvnctionem eivsqve compositionem cvm fvnctione stricte crescente extrema sva in eisdem pvnctis attingere, qvod permittit, exempli gratia, qvadratvm distantiae loco distantiae ipsivs minvere. Deinde demonstrat cviqve pvncto esse pvnctvm proximvm in delineatione fvnctionis in intervallo clavso continvae, et, si hoc in termino non est, pvnctvm datvm in recta normali esse; et ex principio temporis minimi leges refractionis, Snellii, et reflexionis, Heronis, dedvcit. Problemata invenivnt arcam maximi volvminis, cylindrvm minimae svperficiei, vectem longissimvm qvi angvlvm ambvlacri praeterire potest, locvm vnde tabvla svb angvlo maximo conspicitvr et modvm celerrimvm canalem traiciendi.

Proposita discendi

Problema geometricvm vel physicvm in qvaestionem extremi absolvti fvnctionis vnivs variabilis convertere, in dominio accvrate definito.
Ivstificare extremvm inventvm absolvtvm esse: theoremate Weierstrassii, si dominivm est intervallvm clavsvm et circvmscriptvm, et, si non est, signo derivatae, convexitate vel limitibvs in finibvs dominii.
Pvnctvm delineationis pvncto dato proximvm invenire et agnoscere rectam qvae ea conivngit ad delineationem normalem esse.
Ex principio temporis minimi leges reflexionis et refractionis dedvcere, et agnoscere qvando optimvm in termino dominii sit.

Methodvs

Lectiones de theoremate Weierstrassii et de maximis et minimis docvervnt qvomodo extrema absolvta fvnctionis datae in intervallo dato inveniantvr. In problemate optimizationis, contra, optima configvratio figvrae vel motvs qvaeritvr, et fvnctio examinanda, cvm svo dominio, ex envntiato constrvenda est. Methodvs qvinqve articvlos habet.

Variabilis. Qvantitatem eligere qvae configvrationem determinet (longitvdinem, angvlvm, positionem) et per eam, ope condicionvm problematis, omnes ceteras exprimere.
Fvnctio. Qvantitatem qvae maxima vel minima fieri debet vt fvnctionem hvivs vnivs variabilis scribere.
Dominivm. Copiam valorvm qvos variabilis in problemate svmere potest definire, qvae plervmqve intervallvm est. Si fvnctio per continvitatem ad configvrationes degeneres extenditvr, vt rectangvlvm altitvdinis nvllae vel arcam sine volvmine, eas admittere convenit vt intervallvm clavsvm obtineatvr, dvmmodo extremvm qvaesitvm in eis non attingatvr.
Extrema absolvta. Ea in dominio invenire et ivstificare ea absolvta esse. Si dominivm est intervallvm clavsvm et circvmscriptvm, theorema Weierstrassii ea exstare praestat, et propositio de extremis absolvtis in intervallo clavso et circvmscripto, ex lectione de maximis et minimis, ea valoribvs comparandis invenit. Si non est, adhibetvr signvm derivatae (theorema criterii derivatae primae ad extrema absolvta, vel corollarivm de vnico pvncto critico, ex eadem lectione), convexitas (corollarivm de pvnctis criticis fvnctionis convexae, ex lectione de concavitate et convexitate) vel limites in finibvs dominii (propositio de extremis absolvtis in intervallo aperto).
Conclvsio. Ad problema redire: configvrationem optimam dare, non tantvm valorem optimvm, et confirmare eam sensvm habere, in mensvris et in casibvs singvlaribvs.

Qvartvs articvlvs saepissime omittitvr: pvnctvm invenire in qvo derivata evanescit non svfficit. Illvd pvnctvm minimvm esse potest cvm maximvm qvaeritvr, extremvm tantvm locale vel pvnctvm extra dominivm; et extremvm absolvtvm in termino intervalli esse potest, vbi derivata evanescere non debet. Lectio de maximis et minimis ostendit, per $\frac{1}{x} + x^2$ , vnicvm pvnctvm criticvm nihil praestare si dominivm non est intervallvm. Saepe praeterea convenit fvnctionem alia simpliciore svbstitvere qvae extrema sva in eisdem pvnctis attingat.

LemmaExtrema et fvnctiones stricte monotonae

Sint $f$ fvnctio in copia $A$ definita, $c$ pvnctvm ex $A$ et $\varphi$ fvnctio stricte crescens in copia qvae valores ex $f$ continet. Tvm $f$ in $c$ maximvm svvm absolvtvm in $A$ attingit si et solvm si $\varphi \circ f$ id attingit, et idem valet de minimo absolvto, de extremis localibvs et de extremis strictis. Si $\varphi$ est stricte decrescens, $f$ in $c$ maximvm attingit si et solvm si $\varphi \circ f$ in $c$ minimvm eivsdem generis attingit, et vicissim. Fvnctio stricte crescens ordinem inter valores non mvtat, et extrema ab hoc ordine tantvm pendent.

Demonstratio

$u \leq w \Leftrightarrow \varphi(u) \leq \varphi(w), \qquad u < w \Leftrightarrow \dato{orden}{\varphi(u) < \varphi(w)}$
Sint $u$ et $w$ in dominio fvnctionis $\varphi$ . Si $u < w$ , monotonia stricta dat $\varphi(u) < \varphi(w)$ ; si $u = w$ , valores congrvvnt; et si $u > w$ , dat $\varphi(u) > \varphi(w)$ . Ergo inaeqvalitates inter nvmeros et inter eorvm imagines aeqvivalent.
$f(x) \leq f(c) \Leftrightarrow \resaltar{\varphi\left(f(x)\right) \leq \varphi\left(f(c)\right)}$
Cvm $u = f(x)$ et $w = f(c)$ , pro qvoqve $x$ ex $A$ , vel pro qvoqve $x$ ex $A$ qvod a $c$ minvs qvam aliqvod $\delta$ distat, inaeqvalitates qvae maximvm absolvtvm, locale et stricta definivnt, secvndvm lectionem de maximis et minimis et definitionem extremi localis ex lectione de theorematis Fermatii, Rollii, Lagrangii et Cavchii, valent pro $f$ si et solvm si valent pro $\varphi \circ f$ . Minimvm eodem modo tractatvr, et si $\varphi$ est stricte decrescens, inaeqvalitates inter imagines vertvntvr.

Casvs vsitatissimi qvattvor svnt. Radix qvadrata est stricte crescens in $\ico{0}{\infty}$ : distantia eivsqve qvadratvm in eisdem pvnctis minima fivnt, et qvadratvm radicibvs caret. Qvadratvm est stricte crescens in $\ico{0}{\infty}$ : qvantitas non negativa eivsqve qvadratvm in eisdem pvnctis maxima fivnt. Logarithmvs est stricte crescens in $\ioo{0}{\infty}$ : prodvctvm potentiarvm positivarvm eivsqve logarithmvs, qvi svmma est, item. Et fvnctio positiva maximvm svvm attingit vbi eivs inversvm minimvm svvm attingit, qvia $\frac{1}{u}$ est stricte decrescens in $\ioo{0}{\infty}$ .

ExemplvmRectangvlvm semicircvlo inscriptvm

Inter rectangvla semicircvlo radii $R$ inscripta, vno latere in diametro posito, id maximae areae invenire. Ratiocinatio articvlos methodi percvrrit.

Demonstratio

$A(x) = 2x\sqrt{R^2 - x^2}$
Variabilis et fvnctio. Centrvm in origine et diametrvm in axe horizontali ponamvs. Rectangvlvm determinatvr per $x$ , dimidivm basis svae: vertices eivs svperiores in semicircvlo svnt, in $\left(\pm x, \sqrt{R^2 - x^2}\right)$ , et area eivs est haec fvnctio.
$F(x) = \left(A(x)\right)^2 = \resaltar{4x^2\left(R^2 - x^2\right)}, \qquad x \in \icc{0}{R}$
Dominivm. Rectangvla vera respondent valoribvs $x$ in $\ioo{0}{R}$ ; admittimvs etiam $0$ et $R$ , qvae rectangvla degeneria dant, areae nvllae, et intervallvm clavsvm et circvmscriptvm obtinemvs in qvo $A$ continva est: theoremate Weierstrassii maximvm exstat. Cvm $A$ negativa non sit, per lemma svfficit eivs qvadratvm maximvm reddere, qvod polynomivm est et, dissimiliter ac $A$ , etiam in $R$ derivabile est.
$F'(x) = 8x\left(R^2 - 2x^2\right) = 0, \ 0 < x < R \rightarrow \resaltar{x = \frac{R}{\sqrt{2}}}$
Candidata. Derivata fvnctionis $F$ in vno tantvm pvncto intervalli $\ioo{0}{R}$ evanescit.
$F(0) = F(R) = 0 < F\left(\frac{R}{\sqrt{2}}\right) = R^4, \qquad \resaltar{A\left(\frac{R}{\sqrt{2}}\right) = R^2}$
Comparatio et conclvsio. Per propositionem de extremis absolvtis in intervallo clavso et circvmscripto, maximvm fvnctionis $F$ est maximvs valorvm eivs in $0$ , in $R$ et in pvncto critico. Rectangvlvm optimvm basim $\sqrt{2}R$ et altitvdinem $\frac{R}{\sqrt{2}}$ habet, dimidivm basis, et aream $R^2$ .

Rectangvlo optimo ad diametrvm reflexo obtinetvr qvadratvm lateris $\sqrt{2}R$ circvmferentiae integrae inscriptvm, areae $2R^2$ , qvod est, eodem calcvlo, rectangvlvm maximae areae ei inscriptvm. Rectangvla degeneria exitvm non mvtavervnt: maximvm in rectangvlo vero attingitvr.