[{"data":1,"prerenderedAt":180},["ShallowReactive",2],{"topic-page:\u002Fhigh-school\u002Fmathematics-ii\u002Fdifferentiation-and-integration-ideas\u002Fextrema-and-graph-sketching":3},{"topic":4,"category":146,"seo":151,"breadcrumbs":156,"prerequisites":163,"nextTopics":172,"relatedTopics":179},{"id":5,"slug":6,"title":7,"summary":8,"canonicalPath":9,"categoryPath":10,"difficulty":11,"targetAudience":12,"estimatedMinutes":13,"generatedAt":14,"publishedAt":14,"updatedAt":14,"reviewStatus":15,"tags":16,"expectedKnowledge":22,"learningGoal":26,"blocks":27},"high-school-mathematics-ii-differentiation-and-integration-ideas-extrema-and-graph-sketching","extrema-and-graph-sketching","極大・極小とグラフの概形","導関数の符号変化から極大・極小を判定し、増減表と通る点を使って三次関数のグラフの概形を描きます。","\u002Fhigh-school\u002Fmathematics-ii\u002Fdifferentiation-and-integration-ideas\u002Fextrema-and-graph-sketching","\u002Fhigh-school\u002Fmathematics-ii\u002Fdifferentiation-and-integration-ideas","beginner","student_high",6,"2026-07-27","ai_generated",[17,18,19,20,21],"数学II","微分","極大","極小","グラフ",[23,24,25],"導関数の符号","増減表","三次関数","導関数の符号変化から極値を判定して極値を計算し、増減表に基づく三次関数の概形を描けるようになる。",[28,33,43,73,79,89,99,105,111,127,136],{"id":29,"type":30,"title":7,"subtitle":31,"shortDefinition":32},"hero","HeroBlock","増減の向きが切り替わる場所を見る","増加から減少へ変わる点で極大、減少から増加へ変わる点で極小になります。f'=0だけでなく、その左右の符号変化で判定します。",{"id":34,"type":35,"term":36,"definition":37,"plainExplanation":38,"keywords":39},"definition","DefinitionBlock","極値","ある点の近くで関数値が周囲より大きいとき極大値、小さいとき極小値といい、両方を極値といいます。","極値は近くの点との比較です。区間全体で最も大きい最大値・最も小さい最小値とは同じとは限りません。極値を答えるときはx座標だけでなく元の関数値も求めます。",[40,41,42],"極大値","極小値","符号変化",{"id":44,"type":45,"title":46,"formulas":47,"strategy":72},"formula","FormulaBlock","符号変化による極値判定",[48,60],{"label":49,"expression":50,"meaning":51,"symbols":52,"tip":59},"極大の判定","f': 正 → 0 → 負 ⇒ fは極大","増加から減少へ切り替わる頂点型の動き。",[53,56],{"symbol":54,"meaning":55},"f'","元の関数の導関数",{"symbol":57,"meaning":58},"0","接線が水平になる点","増加↗から減少↘へ",{"label":61,"expression":62,"meaning":63,"symbols":64,"tip":71},"極小の判定","f': 負 → 0 → 正 ⇒ fは極小","減少から増加へ切り替わる谷型の動き。",[65,68],{"symbol":66,"meaning":67},"正・負","導関数の左右の符号",{"symbol":69,"meaning":70},"f","元の関数","減少↘から増加↗へ","f'=0の解→左右の符号→極大・極小の種類→元の関数へ代入して極値、の順で判断します。",{"id":74,"type":75,"src":76,"alt":77,"caption":78},"diagram","DiagramBlock","\u002Fassets\u002Fhigh-school\u002Fmathematics-ii\u002Fdifferentiation-and-integration-ideas\u002Fextrema-and-graph-sketching\u002Fextrema-and-graph-sketching.svg","増減表の正負正の符号変化と、対応する三次関数の極大点・極小点を上下に結んだ図","上段の臨界点を下段の(−1,2)、(1,−2)へ垂直に対応させます。曲線は原点を通り、f'の符号どおり↗↘↗と点対称に進みます。",{"id":80,"type":81,"steps":82,"title":88},"steps","StepBlock",[83,84,85,86,87],"f'(x)=0を解き、各区間の符号を調べる","正から負なら極大、負から正なら極小と判定する","候補のxを元のf(x)へ代入して極値を求める","切片など確実に通る点を追加する","増減表の矢印を滑らかな曲線でつなぐ","概形を描く手順",{"id":90,"type":91,"title":92,"points":93,"body":98},"key-points","KeyPointBlock","水平な接線だけでは極値とは限らない",[94,95,96,97],"極大は＋から−への変化","極小は−から＋への変化","極値はf(a)という関数値","概形は増減と通る点を同時に満たす","f'(a)=0は極値の候補を与えますが、左右で符号が変わらなければ極値ではありません。三次関数f(x)=x³ではx=0で接線が水平でも、関数は前後で増加を続けます。",{"id":100,"type":101,"scenario":102,"explanation":103,"result":104},"example","ExampleBlock","f(x)=x³−3xの極値を求め、概形を考える。","f'(x)=3(x−1)(x+1)なので、符号は＋、−、＋です。x=−1で極大、x=1で極小。f(−1)=2、f(1)=−2より、極大値2、極小値−2です。","点(−1,2)で山、点(1,−2)で谷をもち、原点を通る左下から右上への三次曲線になります。",{"id":106,"type":107,"warningTitle":108,"message":109,"severity":106,"action":110},"warning","WarningBlock","極値をx座標だけで答えない","「x=−1で極大」は場所の説明です。極大値を問われたら元の関数へ代入したf(−1)=2まで求めます。","増減表には境界のxと、その真下のf(x)の値を両方書き込み、符号変化の矢印も加えましょう。最後に概形の山と谷が表と一致するか見直します。",{"id":112,"type":113,"title":114,"questions":115},"quiz","QuizBlock","確認テスト",[116],{"question":117,"choices":118,"answerIndex":122,"choiceExplanations":123},"x=aの左右でf'(x)の符号が負から正へ変わります。正しい結論はどれですか。",[119,120,121],"fはx=aで極小になる","fはx=aで極大になる","f(a)=0である",0,[124,125,126],"正解です。減少から増加へ切り替わるため谷型の極小になります。","極大は導関数が正から負へ変わる場合です。","0になると判断できるのは通常f'(a)であり、f(a)の値は元の関数次第です。",{"id":128,"type":129,"title":130,"points":131},"summary","SummaryBlock","まとめ",[132,133,134,135],"極値はf'の左右の符号変化で判定する","正→負は極大、負→正は極小","極値は候補のxを元のfへ代入して求める","増減と通る点を満たすよう概形を描く",{"id":137,"type":138,"title":139,"nextTopics":140,"relatedKeywords":142},"next-action","NextActionBlock","現実の最大・最小問題へ使う",[141],"high-school-mathematics-ii-differentiation-and-integration-ideas-optimization-and-change-modeling",[143,144,145],"最大値","最小値","モデル化",{"path":10,"title":147,"description":148,"parentPath":149,"accentToken":150},"微分・積分の考え","微分係数、導関数、関数の増減と極値、積分、面積を整理していくカテゴリです。","\u002Fhigh-school\u002Fmathematics-ii","signal-blue",{"title":152,"description":153,"canonicalUrl":154,"ogImage":155},"極大・極小とグラフの概形 | 数学II 微分・積分の考え","導関数の符号変化で極大・極小を判定し、増減表から三次関数のグラフの概形を描く方法を学びます。","https:\u002F\u002Fzeqnilo.com\u002Fhigh-school\u002Fmathematics-ii\u002Fdifferentiation-and-integration-ideas\u002Fextrema-and-graph-sketching","https:\u002F\u002Fzeqnilo.com\u002Fassets\u002Fsite\u002Fdefault-og.svg",[157,160,161,162],{"title":158,"path":159},"高校","\u002Fhigh-school",{"title":17,"path":149},{"title":147,"path":10},{"title":7,"path":9},[164],{"id":165,"title":166,"summary":167,"canonicalPath":168,"categoryPath":10,"categoryTitle":147,"difficulty":11,"targetAudience":12,"estimatedMinutes":13,"publishedAt":14,"updatedAt":14,"reviewStatus":15,"tags":169},"high-school-mathematics-ii-differentiation-and-integration-ideas-derivative-sign-and-monotonicity","導関数の符号と関数の増減","導関数が正・負になる区間を符号表で整理し、元の関数が増加する区間と減少する区間を判定します。","\u002Fhigh-school\u002Fmathematics-ii\u002Fdifferentiation-and-integration-ideas\u002Fderivative-sign-and-monotonicity",[17,18,170,171],"増減","符号表",[173],{"id":141,"title":174,"summary":175,"canonicalPath":176,"categoryPath":10,"categoryTitle":147,"difficulty":11,"targetAudience":12,"estimatedMinutes":13,"publishedAt":14,"updatedAt":14,"reviewStatus":15,"tags":177},"微分による最大・最小と変化のモデル化","現実の条件から変数の範囲と目的量の関数を作り、導関数の符号と端点を調べて最大・最小を判断します。","\u002Fhigh-school\u002Fmathematics-ii\u002Fdifferentiation-and-integration-ideas\u002Foptimization-and-change-modeling",[17,18,178,145],"最大最小",[],1785565323403]