@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .
@prefix dc: <http://purl.org/dc/terms/> .
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .

psr: a skos:ConceptScheme .
psr:-K0PQKG10-G
  skos:prefLabel "calcul différentiel"@fr, "differential calculus"@en ;
  a skos:Concept ;
  skos:narrower psr:-PTNR1PV1-4 .

psr:-SKTRS1V0-R
  skos:prefLabel "real analysis"@en, "analyse réelle"@fr ;
  a skos:Concept ;
  skos:narrower psr:-PTNR1PV1-4 .

psr:-PTNR1PV1-4
  skos:broader psr:-SKTRS1V0-R, psr:-K0PQKG10-G ;
  skos:prefLabel "théorème de Fermat sur les points stationnaires"@fr, "Fermat's theorem"@en ;
  dc:modified "2023-08-02"^^xsd:date ;
  skos:definition """En analyse réelle, le théorème de Fermat sur les points stationnaires permet, lors de la recherche d'éventuels extrema locaux d'une fonction dérivable, de limiter l'étude aux zéros de sa dérivée et aux bornes de son ensemble de définition. L'énoncé est le suivant :
         <br/>Soit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle f}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>f</mi>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle f}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="f"></span> une fonction réelle définie sur un intervalle réel <i>ouvert</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle ]a,b[}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mo stretchy="false">]</mo>
         <mi>a</mi>
         <mo>,</mo>
         <mi>b</mi>
         <mo stretchy="false">[</mo>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle ]a,b[}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b51ec208e9582e11a4f340a42d4f17fb4748fcb" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="]a,b["></span> et dérivable en un point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle c\\\\in ]a,b[}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>c</mi>
         <mo>∈<!-- ∈ --></mo>
         <mo stretchy="false">]</mo>
         <mi>a</mi>
         <mo>,</mo>
         <mi>b</mi>
         <mo stretchy="false">[</mo>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle c\\\\in ]a,b[}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a1a9da539d0c4ae16894e231068b7adbd568c66" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.757ex; height:2.843ex;" alt="{\\\\displaystyle c\\\\in ]a,b[}"></span>.
         <br/><i>Si</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle f}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>f</mi>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle f}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="f"></span> possède un extremum local en <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle c}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>c</mi>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle c}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="c"></span>, <i>alors</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle f'(c)=0}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <msup>
         <mi>f</mi>
         <mo>′</mo>
         </msup>
         <mo stretchy="false">(</mo>
         <mi>c</mi>
         <mo stretchy="false">)</mo>
         <mo>=</mo>
         <mn>0</mn>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle f'(c)=0}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f84ce11adae917b3d5f813729260a28227d0e83b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.082ex; height:3.009ex;" alt="{\\\\displaystyle f'(c)=0}"></span>.
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_Fermat_sur_les_points_stationnaires">https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_Fermat_sur_les_points_stationnaires</a>)"""@fr, """In mathematics, Fermat's theorem (also known as interior extremum theorem) is a method to find local maxima and minima of differentiable functions on open sets by showing that every local extremum of the function is a stationary point (the function's derivative is zero at that point). Fermat's theorem is a theorem in real analysis, named after Pierre de Fermat. By using Fermat's theorem, the potential extrema of a function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\displaystyle f}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>f</mi>
         </mstyle>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle \\\\displaystyle f}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07d97d49d29cc57cce35ed58129778c7bb0d8501" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="\\\\displaystyle f"></span>, with derivative <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\displaystyle f'}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mstyle displaystyle="true" scriptlevel="0">
         <msup>
         <mi>f</mi>
         <mo>′</mo>
         </msup>
         </mstyle>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle \\\\displaystyle f'}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c474396362e67718f168a55cf21a864745ac12c" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.005ex; height:2.843ex;" alt="\\\\displaystyle f'"></span>, are found by solving an equation in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\displaystyle f'}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mstyle displaystyle="true" scriptlevel="0">
         <msup>
         <mi>f</mi>
         <mo>′</mo>
         </msup>
         </mstyle>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle \\\\displaystyle f'}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c474396362e67718f168a55cf21a864745ac12c" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.005ex; height:2.843ex;" alt="\\\\displaystyle f'">. Fermat's theorem gives only a necessary condition for extreme function values, as some stationary points are inflection points (not a maximum or minimum). The function's second derivative, if it exists, can sometimes be used to determine whether a stationary point is a maximum or minimum. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Fermat%27s_theorem_(stationary_points)">https://en.wikipedia.org/wiki/Fermat%27s_theorem_(stationary_points)</a>)"""@en ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_Fermat_sur_les_points_stationnaires>, <https://en.wikipedia.org/wiki/Fermat%27s_theorem_(stationary_points)> ;
  skos:inScheme psr: ;
  skos:altLabel "interior extremum theorem"@en ;
  dc:created "2023-08-02"^^xsd:date ;
  a skos:Concept .