Theoremata Fermatii, Rollii, Lagrangii et Cavchii

Svmmarivm

Haec lectio ab eo qvod derivata in pvncto dicit ad id transit qvod in intervallo dicit. Extrema localia definit et theorema Fermatii demonstrat: si fvnctio extremvm locale in pvncto interiore dominii svi attingit ibiqve derivabilis est, derivata eivs evanescit; demonstratio ex propositione de signo derivatae in pvncto, ex lectione de rationibvs mvtationis, oritvr. Vna cvm theoremate Weierstrassii, ex lectione svperiore, Fermativs theorema Rollii praebet; Rollivs, ad differentiam inter fvnctionem et chordam eivs adhibitvs, theorema valoris medii Lagrangii praebet, secvndvm qvod aliqva tangens chordae parallela est; et, ad combinationem dvarvm fvnctionvm adhibitvs, theorema valoris medii Cavchii, qvo lectio proxima regvlam Hospitalii demonstrabit. Ex theoremate valoris medii primae conseqventiae globales orivntvr: fvnctio cvivs derivata in intervallo nvlla est constans est, dvae fvnctiones eadem derivata praeditae constanti differvnt, margo derivatae incrementa circvmscribit, inter dvas radices fvnctionis radix derivatae eivs iacet, et limes derivatae, si exstat, derivata est. Exempla simplicia ostendvnt nvllam hypothesin theorematvm omitti posse.

Proposita discendi

Qvantificatoribvs exprimere qvid sit extremvm locale, et theorema Fermatii demonstrare, condicionem necessariam qvam praebet a condicione svfficienti distingvendo.
Theoremata Rollii, valoris medii Lagrangii et valoris medii Cavchii demonstrare, ea in delineatione legere et exemplis contrariis ostendere hypotheses eorvm omitti non posse.
Ex theoremate valoris medii dedvcere fvnctionem cvivs derivata in intervallo nvlla est constantem esse, et inaeqvalitates marginesqve erroris ex margine derivatae obtinere.
Radices aeqvationis nvmerare, nvmervm earvm svperne theoremate Rollii et inferne theoremate Bolzani circvmscribendo.

Extrema localia et theorema Fermatii

Lectio de theoremate Weierstrassii praestat fvnctionem in intervallo clavso et circvmscripto continvam maximvm et minimvm svvm attingere, sed non dicit vbi. Vt ea locentvr, notio convenit qvae prope pvnctvm tantvm spectet: valor maximvs inter vicinos esse potest, qvamvis non sit maximvs omnivm.

DefinitioExtremvm locale et pvnctvm interivs

Sint $f$ fvnctio in copia $A$ definita et $x_0$ elementvm ipsivs $A$ . Dicitvr $f$ maximvm locale in $x_0$ attingere, si exstat $\delta > 0$ ita vt nvllvs valor ipsivs $f$ in pvnctis ipsivs $A$ qvae a $x_0$ minvs qvam $\delta$ distant valorem $f(x_0)$ svperet, et minimvm locale in $x_0$ attingere, si nvllvs eorvm valorvm minor est qvam $f(x_0)$ :

(\exists \delta > 0)(\forall x \in A)\left(\abs{x - x_0} < \delta \rightarrow f(x) \leq f(x_0)\right), \qquad (\exists \delta > 0)(\forall x \in A)\left(\abs{x - x_0} < \delta \rightarrow f(x_0) \leq f(x)\right)

Extremvm locale est maximvm locale avt minimvm locale. Pvnctvm $x_0$ est pvnctvm interivs ipsivs $A$ , si exstat $r > 0$ ita vt $\ioo{x_0 - r}{x_0 + r} \subseteq A$ .

Si $f$ maximvm svvm in $A$ in pvncto $x_0$ attingit, eo sensv qvo in lectione svperiore, ibi maximvm locale attingit, qvolibet $\delta$ ; propositio conversa falsa est. Consideremvs $f(x) = x^3 - 3x$ . Cvm $2 - f(x) = (x + 1)^2 (2 - x)$ , valet $f(x) \leq 2 = f(-1)$ pro omni $x \leq 2$ : fvnctio in $-1$ maximvm locale attingit, cvm $\delta = 3$ , qvod non est maximvm eivs in $\mathbb{R}$ , qvia $f(3) = 18$ . Pvncta interiora intervalli svnt omnia eivs pvncta praeter terminos qvi ad id pertinent; pvncta interiora ipsivs $\icc{a}{b}$ svnt ea ipsivs $\ioo{a}{b}$ . Extremvm locale in termino iacere potest: fvnctio $x$ , in $\icc{0}{1}$ , minimvm svvm in $0$ et maximvm in $1$ attingit.

TheoremaTheorema Fermatii

Sint $f$ fvnctio in copia $A$ definita et $a$ pvnctvm interivs ipsivs $A$ . Si $f$ in $a$ extremvm locale attingit et in $a$ derivabilis est, tvm $f'(a) = 0$ . Si derivata positiva esset, propositio de signo derivatae ad dextram ipsivs $a$ , proxime, valores maiores qvam $f(a)$ poneret; si negativa esset, eos ad sinistram poneret. Figvra ostendit, cvm $f(x) = x^3 - 3x$ et maximo eivs locali in $a = -1$ , cvr nevtra declivitas fieri possit.

Demonstratio

$x \in A, \ \abs{x - a} < \delta_0 \rightarrow \dato{max}{f(x) \leq f(a)}, \qquad \resaltar{F'(a) = f'(a)}$
Ponamvs primvm $f$ in $a$ maximvm locale attingere, cvm aliqvo $\delta_0 > 0$ , et sit $r > 0$ ita vt intervallvm $I = \ioo{a - r}{a + r}$ in $A$ contineatvr. Sit $F$ restrictio ipsivs $f$ ad $I$ . Omne $x$ ipsivs $A$ cvm $\abs{x - a} < r$ in $I$ iacet; per lemma de indole locali derivatae, ex lectione de regvlis derivandi, $F$ in $a$ derivabilis est, eadem derivata.
$f\left(a + \frac{\rho}{2}\right) > f(a), \qquad \resaltar{f\left(a + \frac{\rho}{2}\right) \leq f(a)}$
Si $f'(a) > 0$ , propositio de signo derivatae in pvncto, ex lectione de rationibvs mvtationis, ad $F$ in intervallo aperto $I$ adhibita, praebet $\delta > 0$ ita vt $f(x) > f(a)$ pro omni $x$ ipsivs $I$ cvm $a < x < a + \delta$ . Sit $\rho$ minimvs nvmerorvm $\delta$ , $\delta_0$ et $r$ . Pvnctvm $a + \frac{\rho}{2}$ in $I$ iacet, hanc condicionem implet et a $a$ minvs qvam $\delta_0$ distat: valor eivs simvl maior qvam $f(a)$ et non maior esset. Ergo $f'(a) > 0$ fieri non potest.
$f\left(a - \frac{\rho}{2}\right) > f(a), \qquad f\left(a - \frac{\rho}{2}\right) \leq f(a) \qquad \rightarrow \qquad \resaltar{f'(a) = 0}$
Si $f'(a) < 0$ , eadem propositio, inaeqvalitatibvs inter valores conversis, praebet $\delta > 0$ ita vt $f(x) > f(a)$ pro omni $x$ ipsivs $I$ cvm $a - \delta < x < a$ , et pvnctvm $a - \frac{\rho}{2}$ eandem contradictionem parit. Cvm derivata neqve positiva neqve negativa sit, nvlla est.
$(-f)'(a) = -f'(a) = 0 \rightarrow \resaltar{f'(a) = 0}$
Si $f$ in $a$ minimvm locale attingit, $-f$ ibi maximvm locale attingit, et in $a$ derivabilis est, derivata $-f'(a)$ , per theorema de derivata combinationis linearis, ex lectione de regvlis derivandi. Ex iis qvae iam demonstrata svnt, haec derivata nvlla est.

Theorema condicionem necessariam praebet, non svfficientem: $x^3$ in $0$ derivatam nvllam habet neqve ibi vllvm extremvm locale attingit, qvia ad sinistram ipsivs $0$ valores negativos et ad dextram positivos capit. Neqve dvae aliae hypotheses omitti possvnt. Si $a$ interivs non est, derivata fortasse non evanescit: fvnctio $x$ , in $\icc{0}{1}$ , maximvm svvm in $1$ attingit, et derivata eivs ibi, qvae est limes a sinistra qvotientis incrementalis qvia dominivm nvlla pvncta ad dextram ipsivs $1$ habet, valet $1$ . Si $f$ in $a$ derivabilis non est, theorema nihil dicit: $\abs{x}$ minimvm svvm in $0$ attingit, vbi derivabilis non est. Ergo extrema localia fvnctionis qvaervntvr inter pvncta interiora in qvibvs derivata evanescit, pvncta in qvibvs derivabilis non est et pvncta qvae interiora non svnt; lectio de monotonia eis nomen dabit et criterivm ad decernendvm qvae extrema sint. Problema solvtvm 1 lectionis de regvlis derivandi animadvertit tangentes horizontales ipsivs $\frac{x}{x^2 + 1}$ delineationem in valoribvs eivs maximo et minimo tangere, et nvntiavit id casv non accidere: fvnctio in $\mathbb{R}$ derivabilis qvae maximvm svvm attingit id in pvncto interiore attingit, vbi theorema Fermatii derivatam evanescere facit.

In reliqva parte hvivs lectionis

Theorema Rollii
Theorema valoris medii Lagrangii
Theorema valoris medii Cavchii
Conseqventiae theorematis valoris medii
Problemata solvta
Exacte dvae radices
Logarithmvs inter dvas fractiones
Solae fvnctiones ratione relativa constanti praeditae
Pvnctvm commvne dvabvs fvnctionibvs
Margo erroris differentialis
Problemata proposita
Valor medivs qvi non exstat
Radices derivatae polynomii
Svmmvm vnvm pvnctvm fixvm
Fvnctiones derivatae svae secvndae oppositae
Pvnctvm Cavchii sinvs et cosinvs