MATHHX B

MATHHX B

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11.1 Funktionsundersøgelse fra første år

Vi starter med at opfriske begreberne fra 1 år. Hvis I har problemer med at regne følgende øvelse så gå tilbage og læs afsnittet om funktioner

Øvelse 11.1.1

Lad \(f(x)=x^3+x^2-2x\). Lav en funktionsundersøgelse af \(f\) ved aflæsning i GeoGebra. I skal altså bestemme:

  • a) Definitionsmængden

  • b) Værdimængden

  • c) Nulpunkter

  • d) Fortegn

  • e) Monotoniforhold

  • f) Ekstrema

Løsning 11.1.1

  • a) \(\textrm {Dm}(f)=\mathbb {R}\)

  • b) \(\textrm {Vm}(f)=\mathbb {R}\)

  • c) \(x=-2\), \(x=0\) og \(x=1\)

  • d) \(f\) er negativ for \(x\in ]-\infty ;-2[\cup ]0;1[\) og \(f\) er positiv for \(x\in ]-2;0[\cup ]1;\infty [\)

  • e) \(f\) er voksende for \(x\in ]-\infty ;-1{,}2]\), voksende for \(x\in [0{,}5;\infty [\) og \(f\) er aftagende for \(x\in [-1{,}2;0{,}5]\).

  • f) \(f\) har lokalt maksimum \(2{,}1\) i \(x=-1{,}2\) og lokalt minimum \(-0{,}63\) i \(x=0{,}5\)

Vi får brug for at lave fortegnsundersøgelser ved beregning (uden at tegne). Så det må vi hellere træne også:

  • Eksempel 11.1.1
    Lad \(f(x)=x^2+x\). Vi vil lave en fortegnsundersøgelse uden at tegne grafen. Vi starter med nulpunkter. Funktionen \(f\) er et andengradspolynomium og vi beregner først diskriminanten:

    \[d=b^2-4ac=1^2-4\cdot 1 \cdot 0=1.\]

    Vi Indsætter nu i nulpunktsformlerne og ser at:

    \[x_1=\frac {-b+\sqrt {d}}{2a}=\frac {-1+\sqrt {1}}{2\cdot 1}=\frac {0}{2}=0\]

    og

    \[x_2=\frac {-b-\sqrt {d}}{2a}=\frac {-1-\sqrt {1}}{2\cdot 1}=\frac {-2}{2}=-1.\]

    Altså \(f\) har nulpunkterne \(x_1=0\) og \(x_2=-1\).

    Nu kan vi finde fortegnsvariationen. Det gør vi ved at lave et sildeben. I sildebenet skal vi bruge begge vores nulpunkter:

    (-tikz- diagram)

    Vi skal også bruge nogle \(x\)-værdier der omgiver vores nulpunkter. Vi bestemmer selv hvilke, men der skal være \(x\)-værdier mellem nulpunkter, og i hver ende også som vist her:

    (-tikz- diagram)

    Vi har altså valgt \(x\)-værdierne \(-2\), \(-0{,}5\) og \(1\). Vi sætter \(x\)-værdierne ind i sildebenet:

    \(\begin {array}{ | c | c | c | c | c |c |} \hline x & -2 & -1 & -0.5 & 0 & 1 \\ \hline f(x) & 2 & 0 & -0.25 & 0 & 2 \\ \hline \end {array}\)

    Ved at kigge på funktionsværdierne kan vi se at \(f\) starter med at være positiv indtil vi rammer første nulpunkt i \(x=-1\), hvorefter den bliver negativ, og så igen positiv efter andet nulpunkt 0. Altså:

    \(f(x)\) er positiv for \(x\in ]-\infty ;-1[\cup ]0;\infty [\)
    \(f(x)\) er negativ for \(x\in ]-1;0[\)
    \(f(x)\) er nul når \(x=-1\) og når \(x=0\).

    Vi husker at \(\in \) betyder ”tilhører” og \(\cup \) betyder de to intervaller til sammen (foreningsmængden).

Øvelse 11.1.2

Bestem med samme metode som i eksempel 11.1.1 en fortegnsundersøgelse for følgende funktioner:

  • a) \(f(x)=x^2-x-6\)

  • b) \(f(x)=2x+8\)

  • c) \(f(x)=x^2\)

  • d) \(f(x)=-7\)

Løsning 11.1.2

  • a) Fortegn for \(f\):
    \(f\) er positiv for \(x\in ]-\infty ;-2[\cup ]3;\infty [\)
    \(f\) er negativ for \(x\in ]-2;3[\)
    \(f\) er nul når \(x=-2\) og når \(x=3\)

  • b) Fortegn for \(f\):
    \(f\) er positiv for \(x\in ]-4;\infty [\)
    \(f\) er negativ for \(x\in ]-\infty ;-4[\)
    \(f\) er nul når \(x=-4\)

  • c) Fortegn for \(f\):
    \(f\) er positiv for \(x\in ]-\infty ;0[\cup ]0;\infty [=\mathbb {R}\setminus \{0\}\)
    \(f\) er nul når \(x=0\)

  • d) Fortegn for \(f\):
    \(f\) er negativ