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Maxima et minima fvnctionis

Svmmarivm

Haec lectio eventvs lectionvm svperiorvm in methodvm colligit qva extrema fvnctionis invenivntvr. Maximvm et minimvm fvnctionis in dominio svo absolvta appellat, vt a localibvs distingvantvr, et extrema stricta definit. Deinde dvo criteria demonstrat qvae decernvnt vtrvm pvnctvm derivata nvlla praeditvm extremvm locale sit, signo derivatae circa illvd non investigato: criterivm derivatae secvndae, qvod ex criterio derivatae primae oritvr, et criterivm derivatarvm svccessivarvm, qvod ex regvla Hospitalii oritvr: si prima derivata qvae in pvncto non evanescit ordinis paris est, extremvm adest, et si ordinis imparis est, non adest. Dimidivm posterivs de extremis absolvtis agit. In intervallo clavso et circvmscripto exstant, per theorema Weierstrassii, et theorema Fermatii ea ad catalogvm candidatorvm redigit, terminos intervalli et pvncta critica, inter qvae satis est valores comparare. In intervallo aperto vel non circvmscripto fieri potest vt non exstent, et de eis monotonia vel limitibvs in terminis intervalli decernitvr; praesertim, si fvnctio in intervallo continva vnicvm pvnctvm criticvm habet et in eo extremvm locale attingit, id extremvm absolvtvm est.

Proposita discendi

  1. Extrema absolvta a localibvs, et stricta ab eis qvae stricta non svnt, qvantificatoribvs distingvere, et qvemqve casvm in exemplis agnoscere.
  2. Criteria derivatae secvndae et derivatarvm svccessivarvm demonstrare, et eis decernere vtrvm pvnctvm derivata nvlla praeditvm maximvm locale, minimvm locale an nevtrvm sit, etiam cvm derivata secvnda evanescit.
  3. Extrema absolvta fvnctionis in intervallo clavso et circvmscripto continvae invenire, valoribvs eivs in terminis intervalli et in pvnctis criticis comparatis.
  4. Decernere vtrvm fvnctio extrema sva absolvta in intervallo aperto vel non circvmscripto attingat, monotonia, vnico pvncto critico vel limitibvs in terminis intervalli.

Extrema absolvta, localia et stricta

Lectio de theoremate Weierstrassii maximvm et minimvm fvnctionis in copia definivit, et lectio de theorematis Fermatii, Rollii, Lagrangii et Cavchii extrema localia, qvae valorem in pvncto cvm valoribvs propinqvis tantvm comparant. Nvnc convenit illis nomen dare qvo ab his distingvantvr, et definire qvando valor in pvncto maior sit omnibvs ceteris cvm qvibvs comparatvr, non solvm maior vel aeqvalis.

DefinitioExtrema absolvta et extrema stricta

Sint ff fvnctio in copia AA definita et x0x_0 elementvm ipsivs AA. Dicitvr ff in x0x_0 maximvm absolvtvm attingere, si in x0x_0 maximvm svvm in AA attingit, et minimvm absolvtvm, si in x0x_0 minimvm svvm in AA attingit, eo sensv qvo in lectione de theoremate Weierstrassii; extremvm absolvtvm est maximvm vel minimvm absolvtvm. Maximvm absolvtvm vel locale strictvm est, si f(x0)f(x_0) maior est qvam qvisqve ceterorvm valorvm cvm qvibvs comparatvr:

(∀x∈A)(x≠x0→f(x)<f(x0)),(∃δ>0)(∀x∈A)(0<∣x−x0∣<δ→f(x)<f(x0))(\forall x \in A)\left(x \neq x_0 \rightarrow f(x) < f(x_0)\right), \qquad (\exists \delta > 0)(\forall x \in A)\left(0 < \abs{x - x_0} < \delta \rightarrow f(x) < f(x_0)\right)

Formvla prima maximvm absolvtvm strictvm definit, secvnda maximvm locale strictvm; minima stricta eodem modo definivntvr, cvm f(x)>f(x0)f(x) > f(x_0).

Extrema absolvta etiam globalia appellantvr, et localia relativa. Omne extremvm absolvtvm locale est, qvolibet δ\delta, et omne extremvm strictvm extremvm est; propositiones conversae falsae svnt. Fvnctio x3−3xx^3 - 3x in −1-1 maximvm locale attingit qvod absolvtvm non est, vt in lectione de theorematis Fermatii, Rollii, Lagrangii et Cavchii visvm est. Cosinvs maximvm svvm absolvtvm in R\mathbb{R} in omni mvltiplo ipsivs 2π2\pi attingit; nvllvm horvm maximorvm absolvtvm strictvm est, qvia valor 11 iteratvr, et qvodqve locale strictvm est, qvia cos⁡x<1\cos x < 1 si 0<∣x−2kπ∣<2π0 < \abs{x - 2k\pi} < 2\pi. Fvnctio constans in omni pvncto maximvm et minimvm absolvta attingit, et nvllvm strictvm est. Vniverse, si ff in x0x_0 simvl maximvm et minimvm locale attingit, prope x0x_0 constans est. Valor maximvs absolvtvs, si exstat, vnicvs est; maximvm absolvtvm strictvm praeterea in vno tantvm pvncto attingitvr.

Extrema a dominio pendent. Si ff extremvm in x0x_0 attingit et BB est svbcopia ipsivs AA qvae x0x_0 continet, restrictio ipsivs ff ad BB qvoqve id attingit, qvia cvm pavcioribvs valoribvs comparatvr; propositio conversa falsa est: x2x^2 in [1,2]\icc{1}{2} minimvm svvm absolvtvm in 11 attingit, et in R\mathbb{R} nvllvm extremvm in 11 habet.